Linear Algebra Scalar and Vector Projection

Question

I have a problem with one part of a question. I just need help with part b. How would I go about applying the equation to find the scalar and vector projection of v2 onto v1? Thanks!

Answer 1

Recall the formulas for scalar and vector projection (respectively), in this case, will be $\frac{\langle v_2,v_1\rangle}{||v_1||}$ and $\frac{\langle v_2,v_1\rangle}{||v_1||^2}v_1$. You have a definition of $\langle\cdot,\cdot\rangle,$ and $||x||^2=\langle x,x\rangle.$ All you need to do is to compute these quantities.

Answer 2

I will extend the previous answer and show how it is computed. As stated, we need to calculate the projection of $v_2$ onto $v_1$. To do this, we use the dot-product, denoted $\langle v_1, v_2 \rangle$. This is simply

$\langle v_1, v_2 \rangle = 2 * 1 + 2 * 0 = 2$.

Now, we project $v_2$ onto $v_1$, denoted $v_2^{||v_1}$, using the formula from previous answer:

$v_2^{||v_1} = \frac{\langle v_1, v_2 \rangle}{\langle v_1, v_1 \rangle} * v_1 = \frac{2}{8} * \begin{pmatrix} 1 \\ 0 \end{pmatrix} = \begin{pmatrix} 0.25 \\ 0 \end{pmatrix}$

Note that $\langle v_1, v_1 \rangle$ is the same as $||v_1||^2$.