Find the vectors whose lengths and directions are given. Try to do the calculations without writing.

Length $\quad \quad$Direction

(d). $a>0 \quad \frac{1}{\sqrt{2}} \mathbf{i}+\frac{1}{\sqrt{3}} \mathbf{j}-\frac{1}{\sqrt{6}} \mathbf{k}$

Find the vectors whose lengths and directions are given. Try to do the calculations without writing.

Length $\quad \quad$Direction

(a). $7 \quad \quad \quad -\mathrm{j}$

Find the distance from the point $(-2,1,4)$ to the

(b). plane $y=-5$

Find the distance from the point (-2, 1, 4) to the a. plane x = 3 b. plane y = -5 c. plane z = -1.

Show that the vector

$$\mathbf { v } = a \mathbf { i } + b \mathbf { j }$$

is parallel to the line bx - ay = c by establishing that the slope of the line segment representing

$$\mathbf { v }$$

is the same as the slope of the given line.

Describe the given set with a single equation or with a pair of equations. The line through the point (1, 3, -1) parallel to the z-axis.

Describe the given set with a single equation or with a pair of equations. The line through the point (1, 3, -1) parallel to the x-axis.

Show that except in degenerate cases,

$$( \mathbf { u } \times \mathbf { v } ) \times \mathbf { w }$$

lies in the plane of

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

whereas

$$\mathbf { u } \times ( \mathbf { v } \times \mathbf { w } )$$

lies in the plane of

$$\mathbf { v } \text { and } \mathbf { w }.$$

What are the degenerate cases?

Describe the given set with a single equation or with a pair of equations. The line through the point (1, 3, -1) parallel to the y-axis.

Parametrize the line segment joining the points P(1, 2, 0) and Q(1, 3, -1).

Sketch the surfaces of the ff Cones.

$$y^2+z^2=x^2$$

Sketch the cone surface in this exercise

$$y^2+z^2=x^2$$

Find a plane through the points P1(1, 2, 3), P2(3, 2, 1) and perpendicular to the plane 4x - y + 2z = 7.

Let $\mathbf { u } , \mathbf { v } , \text { and } \mathbf { w }$ be vectors. Which of the following make sense, and which do not? Give reasons for your answers.

$$\mathbf { u } \cdot ( \mathbf { v } \cdot \mathbf { w } ).$$