91Ó°ÊÓ

For the following exercises, use the given vectors to find the quantities. \(\mathbf{a}=9 \mathbf{i}-2 \mathbf{j}, \mathbf{b}=-3 \mathbf{i}+\mathbf{j}\) a. \(3 \mathbf{a}+\mathbf{b}\) b. \(|\mathbf{a}|\) c. \(\mathbf{a} \times|\mathbf{b} \times| \mathbf{a}\) d. \( \mathbf{b} \times | \mathbf{a}\)

For the following exercises, use the given vectors to find the quantities. Find the values of \(a\) such that vectors \(\langle 2,4, a\rangle\) and \(\langle 0,-1, a\rangle\) and \(\quad\langle 0,-1, a\rangle\) are orthogonal.

For the following exercises, find the unit vectors. Find the unit vector that has the same direction as vector \(\mathbf{v}\) that begins at \((0,-3)\) and ends at \((4,10)\) .

For the following exercises, find the unit vectors. Find the unit vector that has the same direction as vector \(\mathbf{v}\) that begins at \((1,4,10)\) and ends at \((3,0,4) .\)

For the following exercises, find the area or volume of the given shapes. The parallelogram spanned by vectors \(\mathbf{a}=\langle 1,13\rangle\) and \(\mathbf{b}=\langle 3,21\rangle\).

For the following exercises, find the area or volume of the given shapes. The parallelepiped formed by \(\mathbf{a}=\langle 1,4,1\rangle\) and \(\mathbf{b}=\langle 3,6,2\rangle\), and \(\mathbf{c}=\langle- 2,1,-5\rangle\)

For the following exercises, find the vector and parametric equations of the line with the given properties. The line that passes through point \((2,-3,7)\) that is parallel to vector \(\langle 1,3,-2\rangle\)

For the following exercises, find the vector and parametric equations of the line with the given properties. The line that passes through points \((1,3,5)\) and \((-2,6,-3)\)

For the following exercises, find the equation of the plane with the given properties. The plane that passes through point \((4,7,-1)\) and has normal vector \(\mathbf{n}=\langle 3,4,2\rangle\)

For the following exercises, find the equation of the plane with the given properties. The plane that passes through points \((0,1,5),(2,-1,6),\) and \((3,2,5)\)