Question

find the projection of u onto v. u = <-7, -4> v = <-8, -3> proj_v u = write u as the sum of two orthogonal vectors, one of which is proj_v u. u = proj_v u +

Explanation:

Step1: Recall projection formula

The formula for the projection of vector $\mathbf{u}=\langle u_1,u_2
angle$ onto vector $\mathbf{v}=\langle v_1,v_2
angle$ is $\text{proj}_{\mathbf{v}}\mathbf{u}=\frac{\mathbf{u}\cdot\mathbf{v}}{\|\mathbf{v}\|^2}\mathbf{v}$, where $\mathbf{u}\cdot\mathbf{v}=u_1v_1 + u_2v_2$ and $\|\mathbf{v}\|^2=v_1^2 + v_2^2$. Given $\mathbf{u}=\langle - 7,-4
angle$ and $\mathbf{v}=\langle - 8,-3
angle$. First, calculate the dot - product $\mathbf{u}\cdot\mathbf{v}$.
$\mathbf{u}\cdot\mathbf{v}=(-7)\times(-8)+(-4)\times(-3)=56 + 12=68$.

Step2: Calculate the magnitude squared of $\mathbf{v}$

$\|\mathbf{v}\|^2=(-8)^2+(-3)^2=64 + 9=73$.

Step3: Calculate $\text{proj}_{\mathbf{v}}\mathbf{u}$

$\text{proj}_{\mathbf{v}}\mathbf{u}=\frac{\mathbf{u}\cdot\mathbf{v}}{\|\mathbf{v}\|^2}\mathbf{v}=\frac{68}{73}\langle - 8,-3
angle=\langle\frac{68\times(-8)}{73},\frac{68\times(-3)}{73}
angle=\langle-\frac{544}{73},-\frac{204}{73}
angle$.

Step4: Calculate the orthogonal vector

Let $\mathbf{w}=\mathbf{u}-\text{proj}_{\mathbf{v}}\mathbf{u}$.
$\mathbf{w}=\langle - 7,-4
angle-\langle-\frac{544}{73},-\frac{204}{73}
angle=\langle - 7+\frac{544}{73},-4+\frac{204}{73}
angle=\langle\frac{-511 + 544}{73},\frac{-292+204}{73}
angle=\langle\frac{33}{73},-\frac{88}{73}
angle$.

Answer:

$\text{proj}_{\mathbf{v}}\mathbf{u}=\langle-\frac{544}{73},-\frac{204}{73}
angle$; The other orthogonal vector is $\langle\frac{33}{73},-\frac{88}{73}
angle$