Write $\mathbf{u}$ as the sum of a vector parallel to $\mathbf{v}$ and a vector orthogonal to $\mathbf{v}$.

$$\mathbf{u}=3 \mathbf{j}+4 \mathbf{k}, \quad \mathbf{v}=\mathbf{i}+\mathbf{j}$$

Find

$$| \mathbf { v } | , | \mathbf { u } | , \mathbf { v } \cdot \mathbf { u } , \mathbf { u } \cdot \mathbf { v } , \mathbf { v } \times \mathbf { u } , \mathbf { u } \times \mathbf { v }, | \mathbf { v } \times \mathbf { u } |,$$

the angle between

$$\mathbf { v } \text { and } \mathbf { u }$$

the scalar component of

$$\mathbf { u }$$

in the direction of

$$\mathbf { v },$$

and the vector projection of

$$\mathbf { u } \text { onto } \mathbf { v }.$$

$$\left. \begin{array} { l } { \mathbf { v } = \mathbf { i } + \mathbf { j } } \\ { \mathbf { v } = 2 \mathbf { i } + \mathbf { j } - 2 \mathbf { k } } \end{array} \right.$$

Sketch the surface.

$$x ^ { 2 } + 4 z ^ { 2 } = 16$$

Find parametrizations for the line segments joining the points. Draw coordinate axes and sketch each segment, indicating the direction of increasing t for your parametrization. (0, 1, 1), (0, -1, 1)

Show that if $\mathbf{u}, \mathbf{v}, \mathbf{w},$ and $\mathbf{r}$ are any vectors, then

$$(\mathbf{u} \times \mathbf{v}) \cdot(\mathbf{w} \times \mathbf{r})=\left|\begin{array}{ll} \mathbf{u} \cdot \mathbf{w} & \mathbf{v} \cdot \mathbf{w} \\ \mathbf{u} \cdot \mathbf{r} & \mathbf{v} \cdot \mathbf{r} \end{array}\right|.$$

Describe the sets of points in space whose coordinates satisfy the given inequalities or combinations of equations and inequalities.

$$x ^ { 2 } + y ^ { 2 } \leq 1 , \text { no restriction on } z.$$

Show that if $\mathbf{u}, \mathbf{v}, \mathbf{w},$ and $\mathbf{r}$ are any vectors, then

$$\mathbf{u} \times \mathbf{v}=(\mathbf{u} \cdot \mathbf{v} \times \mathbf{i}) \mathbf{i}+(\mathbf{u} \cdot \mathbf{v} \times \mathbf{j}) \mathbf{j}+(\mathbf{u} \cdot \mathbf{v} \times \mathbf{k}) \mathbf{k}$$

Find a unit vector perpendicular to plane PQR. P(1, 1, 1), Q(2, 1, 3), R(3, -1, 1)

Describe the sets of points in space whose coordinates satisfy the given inequalities or combinations of equations and inequalities.

$$x ^ { 2 } + y ^ { 2 } \leq 1 , z = 3$$

Find the area of the triangle determined by the points P, Q, and R. P(1, 1, 1), Q(2, 1, 3), R(3, -1, 1)

Describe the sets of points in space whose coordinates satisfy the given inequalities or combinations of equations and inequalities.

$$x ^ { 2 } + y ^ { 2 } \leq 1 , z = 0$$

Show that if $\mathbf{u}, \mathbf{v}, \mathbf{w},$ and $\mathbf{r}$ are any vectors, then

$$\mathbf{u} \times(\mathbf{v} \times \mathbf{w})+\mathbf{v} \times(\mathbf{w} \times \mathbf{u})+\mathbf{w} \times(\mathbf{u} \times \mathbf{v})=\mathbf{0}$$

Find a vector 5 units long in the direction opposite to the direction of

$$\mathbf { v } = ( 3 / 5 ) \mathbf { i } + ( 4 / 5 ) \mathbf { k }.$$

Find parametrizations for the line segments joining the points. Draw coordinate axes and sketch each segment, indicating the direction of increasing t for your parametrization.