01 Vectors and Geometry in Space Online Quiz Questions

Let $P=(-1,2,5)$ and $Q=(3,4,1)$. Which vector represents the displacement from $P$ to $Q$?

Two vectors with the same magnitude and direction are equal even if their initial points are different.

Find the magnitude of $\mathbf{v}=\langle 3,-4,12\rangle$. Enter the exact numerical value.

For $\mathbf{u}=\langle 2,-1,4\rangle$ and $\mathbf{v}=\langle 1,3,2\rangle$, what can be concluded about the angle between the vectors?

If two vectors in $\mathbb{R}^3$ are parallel, their cross product is the zero vector.

Select all statements that correctly describe the dot product or cross product.

The lines $L_1(t)=(1+t,2t,t)$ and $L_2(s)=(3,4,2)$ intersect. What is the parameter $t$ on $L_1$ at their intersection? Enter the numerical value.

Classify the relationship between the lines $L_1(t)=(t,2t,t)$ and $L_2(s)=(1+2s,2-s,s)$.

Select all valid algebraic tests for the listed geometric relationships.

Find an equation of the plane through the three noncollinear points $P=(1,0,2)$, $Q=(3,1,1)$, and $R=(2,3,3)$. Show how you obtain a normal vector and write the final equation in standard form.

What is the scalar projection of $\mathbf{u}=\langle 2,1,-2\rangle$ onto $\mathbf{v}=\langle 1,2,2\rangle$?

Let $P=(1,-2,3)$ and $Q=(4,5,1)$. Which vector represents the displacement from $P$ to $Q$?

True or false: For vectors $\mathbf{u}$ and $\mathbf{v}$ in $\mathbb{R}^3$, the cross product $\mathbf{u}\times\mathbf{v}$ is perpendicular to both input vectors.

Let $\mathbf{u}=\langle 1,2,-1\rangle$ and $\mathbf{v}=\langle 2,-1,3\rangle$. What type of angle lies between these vectors?

What is the geometric term for a vector perpendicular to every direction lying in a plane?

Which set of parametric equations describes the line through $(2,-1,4)$ with direction vector $\langle 3,0,-2\rangle$?

What is the distance from the point $(1,2,0)$ to the plane $x+2y-2z=3$?

Find the magnitude of the vector $\mathbf{v}=\langle 3,4,12\rangle$. Enter the exact numerical value.