91Ó°ÊÓ

Let \(\mathbf{v}=(a, b, c)\) and \(\mathbf{w}=(3 a, 3 b, 3 c)\) be vectors in \(\mathbb{R}^{3}\). Show that \(\|\mathbf{w}\|=3\|\mathbf{v}\|\).

For Exercises \(5-6,\) find the velocity \(\mathbf{v}(t)\) and acceleration a \((t)\) of an object with the given position vector \(\mathbf{r}(t)\). $$ \mathbf{r}(t)=(t, t-\sin t, 1-\cos t) $$

For Exercises \(3-8,\) find the angle \(\theta\) between the vectors \(\mathbf{v}\) and \(\mathbf{w}\). $$ \mathbf{v}=(2,1,4), \mathbf{w}=(1,-2,0) $$

Find the point(s) of intersection of the sphere \((x-3)^{2}+(y+1)^{2}+(z-3)^{2}=9\) and the line \(x=-1+2 t, y=-2-3 t, z=3+t\)

Write the line \(L\) through the points \(P_{1}\) and \(P_{2}\) in parametric form. \(P_{1}=(1,-2,-3), P_{2}=(3,5,5)\)

For Exercises \(1-6,\) calculate \(\mathbf{v} \times \mathbf{w}\). $$ \mathbf{v}=-\mathbf{i}+2 \mathbf{j}+\mathbf{k}, \mathbf{w}=-3 \mathbf{i}+6 \mathbf{j}+3 \mathbf{k} $$

For Exercises \(5-7,\) write the given equation in (a) cylindrical and (b) spherical coordinates. $$ x^{2}+y^{2}+z^{2}=25 $$

Write the line \(L\) through the points \(P_{1}\) and \(P_{2}\) in parametric form. \(P_{1}=(4,1,5), P_{2}=(-2,1,3)\)

For Exercises \(5-6,\) find the velocity \(\mathbf{v}(t)\) and acceleration a \((t)\) of an object with the given position vector \(\mathbf{r}(t)\). $$ \mathbf{r}(t)=(3 \cos t, 2 \sin t, 1) $$

For Exercises \(1-6,\) calculate \(\mathbf{v} \times \mathbf{w}\). $$ \mathbf{v}=\mathbf{i}, \mathbf{w}=3 \mathbf{i}+2 \mathbf{j}+4 \mathbf{k} $$