Question

Let and be nonzero vectors in the

plane, and let $\theta$ be the angle between them.

a) The component of u along v is the scalar |u| $\cos \theta$ and can be expressed in terms of the dot product as $com{p}_v$ u = __________. Sketch this component in the figure below.

b) The projection of u onto v is the vector $pro{j}_v$u = _______________. Sketch this projection in the figure below.

Short answer

a. The component of $\overrightarrow{u}$along $\overrightarrow{v}$is the scalar $|\overrightarrow{u} |\cos \theta$and can be expressed in terms of the dot product as $com{p}_{p_{v}u}=\frac{\overrightarrow{u} ·\overrightarrow{v} }{|\overrightarrow{v} |}$.

b. The projection of $\overrightarrow{u}$onto $\overrightarrow{v}$ is $\mathrm{Pr}o{j}_{v}u=\left(\frac{\overrightarrow{u} \overrightarrow{v} }{|\overrightarrow{v} {|}^2}\right)\overrightarrow{v}$

Step-by-step solution

a.Step 1. Given information.

The component of$\overrightarrow{u}$ along $\overrightarrow{v}$.

Step 2. Determining the dot product.

Given the component of $\overrightarrow{u}$along $\overrightarrow{v}$.

Then

$com{p}_{p_{v}u}=\frac{\overrightarrow{u} ·\overrightarrow{v} }{|\overrightarrow{v} |}$

b.Step 1. Given information.

The projection of $\overrightarrow{u}$ onto $\overrightarrow{v}$.

Step 2. Determining the dot product.

Given the projection of $\overrightarrow{u}$onto $\overrightarrow{v}$.

Then

$pro{j}_{v}u=\left(\frac{\overrightarrow{u} ·\overrightarrow{v} }{|\overrightarrow{v} {|}^2}\right)\overrightarrow{v}$