Vector
- What Is a Vector?
- Vectors in Euclidean Geometry - Definition
- Representation of Vectors
- Components of a Vector
- Magnitude of Vectors
- Angle Between Two Vectors
- Types of Vectors
- Properties of Vectors
- Vector Addition
- Vector Subtraction
- Scalar Multiplication of Vectors
- Multiplication of Vectors
- Scalar Triple Product of Vectors
- Scalars and Vectors
- Applications of Vectors
- Solved Examples
- Practice Problems
- Frequently Asked Questions
What Is a Vector?
Vectors are quantities that have both magnitude and direction. They represent quantities like displacement, velocity, or force that not only tell you how much of something there is but also in which direction it acts.
For instance, if you're moving an object from point `A` to point `B`, the displacement of the object can be represented as a vector pointing from `A` to `B`. Here point `A` would act as the tail of the vector and `B` as the head. The length of `AB` signifies the magnitude of the vector.
Vectors in Euclidean Geometry - Definition
In Euclidean geometry, vectors are quantities characterized by both magnitude and direction. These mathematical entities play a crucial role in understanding spatial relationships and transformations. A vector in Euclidean space can be represented by an ordered pair of numbers, typically denoted as `(x, y)` in a two-dimensional plane or `(x, y, z)` in three-dimensional space. These numbers correspond to the coordinates of a point relative to a chosen origin. Vectors can be added together or multiplied by scalars, allowing for operations like translation, rotation, and scaling in geometry. They are fundamental in describing lines, planes, and shapes and in solving various geometric problems.
Representation of Vectors
Vectors are typically depicted using bold lowercase letters, such as a, or with an arrow over the letter, like `vec{a}`. Another common notation involves indicating vectors by their starting and ending points with an arrow above them; for instance, vector `AB` can be represented as `vec{AB}`. The standard form of representing a vector is `vec{A} = ahat{i} + bhat{j} + chat{k}`, where `a`, `b`, and `c` are real numbers, and `hat{i}`, `hat{j}`, and `hat{k}` are unit vectors along the `x`-axis, `y`-axis, and `z`-axis respectively.
Vectors, symbolized by arrows, have two main points: the initial point, known as the tail, and the terminal point, referred to as the head. They characterize the displacement or movement of an object from one location to another.
In the figure below, the length of the line `AB` is the magnitude, and the head of the arrow points toward the direction of displacement. The vector between `A` and `B` is written as `vec{AB}` or `vec{a}`. The arrow over the letters `AB` signifies the direction of the vector from `A` to `B`.
In a Cartesian coordinate system, vectors can be also expressed using ordered pairs.
Components of a Vector
A vector possesses two fundamental characteristics: magnitude and direction. When comparing two vector quantities of the same type, both magnitude and direction are essential considerations. In a two-dimensional coordinate system, any vector can be deconstructed into its `x`-component, denoted as \( V_x \), and its `y`-component, denoted as \( V_y \).
To calculate the values of \( V_x \) and \( V_y \), the following formulas are utilized:
\( V_x = V \cdot \cos\theta \)
\( V_y = V \cdot \sin\theta \)
The magnitude of the vector \( |V| \) can be determined using the Pythagorean theorem, which states:
\( |V| = \sqrt{V_x^2 + V_y^2} \)
Example: Determine the `x` and `y` components of the vector shown below.
Solution:
\( V_x = V \cdot \cos\theta \)
Substituting the value of \( \theta \) `= 60°` and `|V| = 15`, we get
\( V_x = 15 \cdot \cos 60° \)
\( V_x = 15 \cdot 0.5 \)
\( V_x = 7.5 \)
Similarly, \( V_y = V \cdot \sin\theta \)
Substituting the value of \( \theta \) `= 60°` and `|V| = 15`, we get
\( V_x = 15 \cdot \sin 60° \)
\( V_x = 15 \cdot 0.87 \)
\( V_x = 13.05 \)
Hence, the `x` and `y` components of the vector `V` are \( V_x = 7.5 \) and \( V_x = 13.05 \).
Magnitude of a vector
To determine the magnitude of a vector, you use a simple formula: take the square root of the sum of the squares of its components.
For a vector `vec{A}` with components `(x, y)`, the magnitude formula is expressed as:
`|A| = sqrt(x^2 + y^2)`
Likewise, for a vector `vec{A}` with components `(x, y, z)`, the magnitude formula is expressed as:
`|A| = sqrt(x^2 + y^2 + z^2)`
This formula calculates the length or size of the vector, yielding a scalar value that represents its magnitude.
Angle between two vectors
To find the angle between two vectors, you utilize the dot product formula. Let's take two vectors, `a` and `b`, and denote the angle between them as `θ`. The dot product of these vectors is given by `a·b = |a||b| cosθ`.
Our goal is to determine the angle `θ`. This angle not only signifies the separation between the vectors but also indicates their relative directions. The value of `θ` can be determined using the following formula:
`θ = cos^(-1)[(a·b)/(|a||b|)]`
Types of Vectors
Vectors are classified into different types based on their characteristics such as magnitude, direction, and relationships with other vectors. Let's explore several types of vectors and their properties:
Zero Vectors:
Vectors with zero magnitude are termed as zero vectors, symbolized as `vec{0} = (0,0,0)`. They possess no direction and are considered the additive identity of vectors.
Example: Find the zero vector in `4`-dimensional space.
Solution: The zero vector in `4`-dimensional space is represented as `vec{0} = (0, 0, 0, 0)`. It has zero magnitude and no direction, serving as the additive identity in vector operations.
Unit Vectors:
Unit vector has a magnitude of `1` and unit vector notation is `hat{a}`. They serve as the multiplicative identity of vectors, indicating direction, with a fixed magnitude of `1`.
Example: Determine the unit vector in the direction of the vector `vec{v} = (3, 4, 5)`.
Solution:
First, find the magnitude of `vec{v}` using the formula: `|vec{v}| = sqrt(3^2 + 4^2 + 5^2) = sqrt(9 + 16 + 25) = sqrt50`.
Then, divide each component of `vec{v}` by its magnitude to obtain the unit vector: `hat{a} = (3/sqrt50, 4/sqrt50, 5/sqrt50)`.
Position Vectors:
These vectors determine the position and direction of movement in three-dimensional space. Their magnitude and direction can vary relative to other bodies, hence also referred to as location vectors.
Example: Given a point `P` with coordinates `(2, -3, 1)`, find the position vector from the origin to point `P`.
Solution:
The position vector `vec{OP}` is determined by the coordinates of point `P` relative to the origin `O`: `vec{OP} = (2, -3, 1)`.
Equal Vectors:
Two or more vectors are equal if their corresponding components are identical, sharing the same magnitude and direction. They may have different initial and terminal points, however, their overall characteristics remain the same.
Example: Are the vectors `vec{a} = (1, 2, 3)` and `vec{b} = (1, 2, 3)` equal?
Solution: Yes, the vectors `vec{a}` and `vec{b}` are equal since their corresponding components are identical, indicating the same magnitude and direction.
Negative Vector:
A vector is considered the negative of another if it possesses equal magnitudes but opposite directions. If vectors `A` and `B` have equivalent magnitudes but opposing directions, `A` is regarded as the negative of `B`, or vice versa.
Example: If `vec{a} = (2, -4, 3)`, find the negative of vector `vec{a}`.
Solution:
The negative of the vector `vec{a}` is obtained by changing the sign of each component: `-vec{a} = (-2, 4, -3)`.
Parallel Vectors:
Parallel vectors share the same direction but may have differing magnitudes. Their direction angles coincide, with a difference of zero degrees. Vectors whose direction angles differ by `180` degrees are antiparallel, signifying opposite directions.
Example: Determine if the vectors `vec{p}``= (3, -1, 2)` and `vec{q}` `= (-6, 2, -4)` are parallel.
Solution:
Since both vectors have proportional components, they are parallel. To verify, calculate the ratios of corresponding components: `(p_1)/(q_1) = 3/-6 = -1/2`, `(p_2)/(q_2) = -1/2 = 1/2`, `(p_3)/(q_3) = 2/-4 = -1/2`.
Orthogonal Vectors:
Vectors are orthogonal if the angle between them measures `90` degrees. This property implies that their dot product always equals zero, indicating perpendicularity.
Example: Are the vectors `vec{x}` `= (1, 0, -1)` and `vec{y}` `= (1, 2, 1)` orthogonal?
Solution:
To determine orthogonality, compute the dot product of `vec{x}` and `vec{y}`: `vec{x}·vec{y} = (1)(1) + (0)(2) + (-1)(1) = 1 + 0 - 1 = 0`. Since the dot product is zero, the vectors are orthogonal.
Co-initial Vectors:
Vectors sharing the same initial point are termed co-initial vectors, representing a common starting position.
Example: Given two vectors `vec{u}` `= (2, 3, -1)` and `vec{v}` `= (-1, 2, 4)`, are they co-initial?
Solution:
Vectors are co-initial if they share the same initial point. Here, `vec{u}` and `vec{v}` do not share the same initial point, hence they are not co-initial.
Properties of Vectors
Commutative and Associative Properties of Addition:
- The addition of vectors is commutative, meaning the order of addition doesn't affect the result.
- It is also associative, meaning the grouping of added vectors doesn't change the outcome.
Dot Product Property:
The dot product of two vectors results in a scalar quantity and lies within the plane formed by the two vectors.
Cross-Product Property:
The cross product of two vectors yields a vector perpendicular to the plane containing those two vectors.
Orthogonality and Parallelism:
- Unit vectors (`hat{i}`,`hat{j}`, `hat{k}`) are orthogonal to each other and have a dot product of zero.
- The cross product of two parallel vectors is zero.
Vector Addition
Vector addition mirrors scalar addition, where the individual components of respective vectors are combined to yield the final result:
`vec{a} + vec{b} = (a_1 hat{i} + a_2 hat{j} + a_3 hat{k}) + (b_1 hat{i} + b_2 hat{j} + b_3 hat{k})`
`= (a_1, a_2, a_3) + (b_1, b_2, b_3)`
`= (a_1 + b_1, a_2 + b_2, a_3 + b_3)`
`= (a_1 + b_1) hat{i} + (a_2 + b_2) hat{j} + (a_3 + b_3) hat{k}`
Vector addition exhibits two fundamental laws:
Triangle Law of Vector Addition:
The Triangle Law of Vector Addition simplifies the process of adding vectors. It suggests that if two consecutive sides of a triangle represent two vectors, their sum or resultant vector is represented by the third side of the triangle taken in the opposite direction, which completes the geometric figure. In simpler terms, if you have two vectors acting on an object in sequence, the total effect of both vectors is as if they acted together in a straight line from the starting point to the ending point.
Example: A woman walks `2.5` miles east then turns and walks `6.9` miles north. Where is she with respect to her starting point?
Solution:
First let’s calculate the diagonal distance between the starting and the end point. Here we can use the triangle law of vector addition to find the magnitude of the resultant vector.
The magnitude of the resultant vector `= sqrt(2.5^2+6.9^2) = sqrt(2.5^2+6.9^2) = sqrt53.86 = 7.34` miles
Let’s now calculate the angle of the resultant vector with respect to the starting point.
The angle of the resultant vector `= x° = Tan^-1 (6.9/2.5) = 70°` north of east.
Therefore, with respect to her starting point, the woman is at a distance of `7.34` miles away in the direction `70°` north of east.
Parallelogram Law of Vector Addition:
The Parallelogram Law of Vector Addition is another straightforward concept. It states that if two vectors are along the adjacent sides of a parallelogram, the diagonal starting from the point of contact of the two vectors represents their sum or resultant vector. In essence, when two vectors act simultaneously from a common starting point, their combined effect is represented by the diagonal of the parallelogram formed by the vectors. This law simplifies the process of finding the resultant of two vectors graphically.
Example: In the diagram given below are the magnitude and the direction of two vectors. Sketch the resultant vector and calculate its magnitude and direction.
Solution:
Vector Subtraction
Subtracting one vector from another involves subtracting their respective components:
`(a_1 hat{i} + a_2 hat{j} + a_3 hat{k}) - (b_1 hat{i} + b_2 hat{j} + b_3 hat{k}) = (a_1 - b_1) hat{i} + (a_2 - b_2) hat{j} + (a_3 - b_3) hat{k}`
Scalar Multiplication of Vectors
Scalar multiplication of vectors involves multiplying each component of the vector by a scalar, which is simply a real number with no direction. This operation alters the magnitude of the vector without changing its direction. When a vector `a = (a_1, a_2, a_3) = a_1 hat{i} + a_2 hat{j} + a_3 hat{k}` is multiplied by a scalar `r`, the resulting vector is:
`ra = (ra_1, ra_2, ra_3) = (ra_1)hat{i} + (ra_2)hat{j} + (ra_3)hat{k}`
If the scalar `r` is negative, the direction of a vector reverses by `180` degrees.
- Scalar multiplication exhibits several key properties:
- Scaling a vector by a constant `k` distributes over vector addition: `k(vec{a} + vec{b}) = kvec{a} + kvec{b}`
- Distributive property also holds when a vector is multiplied by the sum of two scalars: `(k + l)vec{a} = kvec{a} + lvec{a}`
- The dot product of a vector with the scalar `1` remains unchanged: `vec{a}·1 = vec{a}`
- Multiplying a vector by the scalar `0` results in the zero vector: `vec{a}·0 = vec{0}`
- Multiplying a vector by the scalar `-1` reverses its direction: `vec{a}·(-1) = vec{-a}`
Multiplication of Vectors
Vector multiplication operates differently from real number multiplication. There are two primary methods for multiplying vectors:
Dot Product of Vectors
In vector algebra, the components of the two vectors are multiplied individually and then summed to yield the dot product. Mathematically, this is represented as:
\( \text{a·b} = (a_1 \hat{i} + a_2 \hat{j} + a_3 \hat{k}) \cdot (b_1 \hat{i} + b_2 \hat{j} + b_3 \hat{k}) = (a_1, a_2, a_3) \cdot (b_1, b_2, b_3) = (a_1 \cdot b_1) + (a_2 \cdot b_2) + (a_3 \cdot b_3) \)
Where `vec{a} = (a_1 hat{i} + a_2 hat{j} + a_3 hat{k})` and `vec{b} = (b_1 hat{i} + b_2 hat{j} + b_3 hat{k})`
Another approach to finding the dot product involves multiplying the magnitudes of the vectors and the cosine of the angle between them:
\( \vec{A} \cdot \vec{B} = |\vec{A}| |\vec{B}| \cos \theta \)
The dot product yields a scalar value and does not include any direction.
Cross Product of Vectors
Cross-product multiplication involves representing the vector components in a matrix and evaluating the determinant of the matrix to determine the result. Mathematically, it is expressed as:
\( \text{A × B} = (a_1 \hat{i} + a_2 \hat{j} + a_3 \hat{k}) × (b_1 \hat{i} + b_2 \hat{j} + b_3 \hat{k}) = (a_2b_3 - a_3b_2, a_3b_1 - a_1b_3, a_1b_2 - a_2b_1) \)
Alternatively, the cross product can be computed by multiplying the magnitudes of the vectors and the sine of the angle between them, with a direction given by the unit normal vector:
\( \vec{A} \times \vec{B} = |\vec{A}| |\vec{B}| \sin \theta\hat{n} \)
`|A|` is the magnitude of the vector `vec{A}`
`|B|` is the magnitude of the vector `vec{B}`
`θ` is the angle between `vec{A}` and `vec{B}`
`n` is the unit vector at right angles to both `vec{A}` and `vec{B}`
Scalar Triple Product of Vectors
The scalar triple product of vectors involves taking the dot product of one vector with the cross product of the other two vectors. If any two vectors in a scalar triple product are identical, then the scalar triple product equals zero. Furthermore, if the scalar triple product equals zero, the three vectors, namely `a`, `b`, and `c` lie in the same plane.
Additionally, it's important to note that the scalar triple product has a cyclic property, meaning that:
`a·(b × c) = b·(c × a) = c·(a × b)`
Scalars and Vectors
Scalars
Scalars are physical quantities devoid of direction. They are represented solely by magnitude and are often accompanied by unit measurements. Examples of scalars include time, temperature, mass, and energy. Basic mathematical operations such as addition, subtraction, multiplication, and division apply to scalars.
For example, a temperature of `25°C` or a mass of `5` kilograms are scalar quantities as they represent magnitude without direction.
Vectors
Vectors, on the other hand, are physical quantities possessing both magnitude and direction. They represent quantities like displacement, velocity, force, and acceleration. Unlike scalars, vectors require both magnitude and direction to be fully defined. Vectors can undergo basic mathematical operations like addition, subtraction, and multiplication, but their properties differ from those of scalars due to the consideration of direction.
For example, a displacement of `10` meters east indicates movement in the eastward direction, while a velocity of `20` meters per second north signifies motion in the northward direction.
Applications of Vectors
Vectors find extensive applications in both Physics and Mathematics, playing a crucial role in various fields. Some of the key applications of vectors include:
`1`. Representation of Physical Quantities
Vectors are instrumental in representing the position, displacement, velocity, acceleration, etc. of objects and physical quantities. They provide a concise and precise way to describe the direction and magnitude of these quantities.
`2`. Mathematical Analysis
Vectors are fundamental in the study of partial differential equations and differential geometry. They are utilized to analyze complex mathematical models and systems, facilitating the understanding of geometric and analytical concepts.
`3`. Physics and Engineering
In physics and engineering, vectors are extensively employed, particularly in the analysis of electromagnetic fields, gravitational fields, and fluid flow. They enable scientists and engineers to model and predict the behavior of physical phenomena accurately.
Solved Examples
Example `1`. Given two vectors `A = (3, 4)` and `B = (-2, 6)`, find the resultant vector when they are added together.
Solution:
To find the resultant vector, simply add the corresponding components of the two vectors.
`A + B = (3 + (-2), 4 + 6) = (1, 10)`
So, the resultant vector is `R = (1, 10)`.
Example `2`. Given a vector `V = (2, -5)` and a scalar value \( k = 3 \), find the result of scalar multiplication \( kV \).
Solution:
To perform scalar multiplication, multiply each component of the vector by the scalar.
\( kV \) = \( 3 \) `* (2, -5) = (3*2, 3*(-5)) = (6, -15)`
So, the result of the scalar multiplication is \( (6, -15) \).
Example `3`. Calculate the dot product of the vectors `P = (2, 3)` and `Q = (5, -1)`.
Solution:
The dot product of two vectors is found by multiplying the corresponding components of the vectors and then summing the results.
`P · Q = (2 * 5) + (3 * -1) = 10 - 3 = 7`
So, the dot product of `P` and `Q` is `7`. This is a scalar quantity.
Example `4`. Find the cross product of the vectors `X = (1, 2, -3)` and `Y = (4, -5, 6)`.
Solution:
The cross-product of two vectors in three-dimensional space is calculated using the determinant method.
\( \text{X × Y} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 1 & 2 & -3 \\ 4 & -5 & 6 \end{vmatrix} \)
\( = \hat{i}(2×6 - (-3)×(-5)) - \hat{j}(1×6 - (-3)×4) + \hat{k}(1×(-5) - 2×4) \)
\( = \hat{i}(12 - 15) - \hat{j}(6 + 12) + \hat{k}(-5 - 8) \)
\( = \hat{i}(-3) - \hat{j}(18) + \hat{k}(-13) \)
So, the cross product of `X` and `Y` is \( (-3, -18, -13) \). This is a vector quantity.
Example `5`. An object moves from point `A` to point `B` with a displacement vector `d = (5, 3)` meters. Calculate the magnitude of the displacement and the angle it makes with the `x`-axis.
Solution:
To find the magnitude of the displacement, use the Pythagorean theorem:
\( |d| = \sqrt{(5)^2 + (3)^2} = \sqrt{25 + 9} = \sqrt{34} \)
To find the angle \( \theta \) with the `x`-axis, use trigonometric functions:
\( \tan\theta = \frac{3}{5} \)
\( \theta = \arctan\left(\frac{3}{5}\right) \)
\( \theta \approx 30.96^\circ \)
So, the magnitude of displacement is \( \sqrt{34} \) meters and the angle with the x-axis is approximately \( 30.96^\circ \).
Practice Problems
Q`1`. Given two vectors `A = (2, -3)` and `B = (-1, 5)` what is the resultant vector when they are added together?
- `(1, 2)`
- `(3, 2)`
- `(1, 8)`
- `(-3, 2)`
Answer: a
Q`2`. If vector `V = (-3, 4)` and the scalar value \( k = -2 \), what is the result of scalar multiplication \( kV \)?
- `(6, -8)`
- `(-6, 8)`
- `(-3, -4)`
- `(3, 4)`
Answer: a
Q`3`. Compute the dot product of the vectors `P = (3, -2)` and `Q = (1, 6)`.
- `15`
- `-15`
- `18`
- `-9`
Answer: d
Q`4`. Find the cross product of the vectors `X = (2, 1, -3)` and `Y = (-1, 4, 2)`.
- `(10, -8, 9)`
- `(14, -1, 9)`
- `(-14, 1, -9)`
- `(-10, -8, -9)`
Answer: b
Q`5`. An object moves from point `A` to point `B` with a displacement vector `d = (6, -8)` meters. Calculate the magnitude of the displacement.
- `10` meters
- `5` meters
- `14` meters
- `18` meters
Answer: a
Frequently Asked Questions
Q`1`. What are scalars and vectors?
Answer: Scalars are quantities with only magnitude, while vectors have both magnitude and direction.
Q`2`. How are vectors represented in mathematics?
Answer: Vectors can be represented geometrically by arrows or mathematically as ordered pairs or triples.
Q`3`. What is the difference between dot product and cross product?
Answer: The dot product yields a scalar quantity, while the cross product results in a vector.
Q`4`. How are vectors used in physics?
Answer: Vectors are used to describe physical quantities such as force, velocity, and acceleration, which have both magnitude and direction.
Q`5`. What is the importance of vector addition?
Answer: Vector addition is crucial for determining the resultant vector when multiple vectors act simultaneously.
Q`6`. Can vectors be multiplied by scalars?
Answer: Yes, vectors can be multiplied by scalars, resulting in a scalar multiple of the original vector.
Q`7`. How are vectors applied in engineering?
Answer: Vectors are widely used in engineering for analyzing forces, motion, and fluid dynamics, among other applications.