Linear Algebra with Applications, 10th Edition

Section 5.1Exercises

1. Find the angle between the vectors v and w in each of the following:

   1. $\mathbf{v}={(2,1,3)}^T,\mathbf{w}={(6,3,9)}^T$

   2. $\mathbf{v}={(2,-3)}^T,\mathbf{w}={(3,2)}^T$

   3. $\mathbf{v}={(4,1)}^T,\mathbf{w}={(3,2)}^T$

   4. $\mathbf{v}={(-2,3,1)}^T,\mathbf{w}={(1,2,4)}^T$

2. For each pair of vectors in Exercise 1, find the scalar projection of v onto w. Also find the vector projection of v onto w.

3. For each of the following pairs of vectors x and y, find the vector projection p of x onto y and verify that p and $\mathbf{x}-\mathbf{p}$ are orthogonal:

   1. $\mathbf{x}={(3,4)}^T,\mathbf{y}={(1,0)}^T$

   2. $\mathbf{x}={(3,5)}^T,\mathbf{y}={(1,1)}^T$

   3. $\mathbf{x}={(2,4,3)}^T,\mathbf{y}={(1,1,1)}^T$

   4. $\mathbf{x}={(2,-5,4)}^T,\mathbf{y}={(1,2,-1)}^T$

4. Let x and y be linearly independent vectors in ${ℝ}^2$. If $\Vert \mathbf{x}\Vert =2$ and $\Vert \mathbf{y}\Vert =3$, what, if anything, can we conclude about the possible values of $\left|{\mathbf{x}}^T\mathbf{y}\right|$?

5. Find the point on the line $y=2x$ that is closest to the point (5, 2).

6. Find the point ...