Unit 1 test: Vectors and 3D Space

Find the distance between $(2, -1, 3)$ and $(-4, 1, 6)$.

Find the radius of the sphere $x^2 + y^2 + z^2 - 6x + 4z = 12$.

Find the unit vector in the direction of $\langle -2, 6, 3 \rangle$.

Let $\mathbf{a} = \langle 1, -3, 4 \rangle$ and $\mathbf{b} = \langle 5, 2, -1 \rangle$. Find $2\mathbf{a} - \mathbf{b}$.

Find the angle between $\langle 1, -1, 2 \rangle$ and $\langle 2, 1, -1 \rangle$, in degrees, rounded to the nearest tenth.

Find the value of $c$ for which $\langle 2, c, -1 \rangle$ and $\langle c, 3, 8 \rangle$ are orthogonal.

Find the vector projection of $\mathbf{b} = \langle 6, -3, 3 \rangle$ onto $\mathbf{a} = \langle 2, 1, -2 \rangle$.

Compute $\langle 2, 0, -1 \rangle \times \langle 1, 3, 2 \rangle$.

Find the area of the triangle with vertices $P(1, 0, 0)$, $Q(0, 2, 0)$ and $R(0, 0, 3)$.

Find the volume of the parallelepiped with edges $\langle 1, 0, 2 \rangle$, $\langle 0, 3, 1 \rangle$ and $\langle 2, 1, 0 \rangle$.

The plane through $(1, -1, 2)$ perpendicular to the line $x = 3t,\ y = 1 - t,\ z = 2 + 2t$ has equation $3x - y + 2z = d$. Find $d$.

Find the point where the line $x = 2 - t,\ y = 1 + 3t,\ z = 4t$ meets the plane $2x + y - z = 7$.

Classify the lines $L_1$: $x = 1 + t,\ y = 1 + 2t,\ z = 3 - t$ and $L_2$: $x = 3 + s,\ y = 4 + s,\ z = 3 + s$.

Find the distance from the origin to the plane $2x - 3y + 6z = 14$.

Identify the surface $x^2 + y^2 - 9z^2 = 9$.