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Let $u=-5i+2j$ and $v=-5i-5j$ . Decompose $u$ into two vectors $u_{1}$ and $u_{2}$ , where $u_{1}$ is parallel to $v$ and $u_{2}$ is orthogonal to $v$ . Write your answers in the form $ai+bj$ .

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Write equations of sine and cosine functions using properties

Full solution

Q. Let

u=-5i+2j

and

v=-5i-5j

. Decompose

u

into two vectors

u_{1}

and

u_{2}

, where

u_{1}

is parallel to

v

and

u_{2}

is orthogonal to

v

. Write your answers in the form

ai+bj

.

Identify vectors $u$ and $v$ : Identify the vectors $u$ and $v$ . $u = -5i + 2j$ $v = -5i - 5j$
Calculate projection of $u$ : Calculate the projection of $u$ onto $v$ to find $u_{1}$ , which is parallel to $v$ . Use the formula: $\text{proj}_v(u) = \frac{u \cdot v}{v \cdot v} \times v$ First, find $u \cdot v$ : $u \cdot v = (-5)(-5) + (2)(-5) = 25 - 10 = 15$ Then, find $v \cdot v$ : $v \cdot v = (-5)(-5) + (-5)(-5) = 25 + 25 = 50$ Now, calculate the projection: $u$ $0$
Calculate $u_{2}$ : Calculate $u_{2}$ , which is orthogonal to $v$ . Use the formula: $u_{2} = u - u_{1}$ $u_{2} = (-5i + 2j) - (-1.5i - 1.5j) = -3.5i + 3.5j$

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