1 Vectors and Geometry Online Quiz Questions

Let $\mathbf u=\langle 2,-1,3\rangle$ and $\mathbf v=\langle -1,4,2\rangle$. Which vector equals $2\mathbf u-\mathbf v$?

True or false: The Euclidean norm of $\langle 3,-4,12\rangle$ is $13$.

Find the Euclidean norm of $\langle 2,-3,6\rangle$. Enter the exact numerical value.

Select all statements that are correct.

Which vector equation describes the line through $P=(1,-2,3)$ and $Q=(5,1,0)$?

What is the standard term for the vector component of $\mathbf v$ parallel to a nonzero vector $\mathbf u$, given by $\frac{\mathbf v\cdot\mathbf u}{\|\mathbf u\|^2}\mathbf u$? Enter the term.

True or false: The span of one nonzero vector in $\mathbb R^3$ is a plane through the origin.

Find an equation of the plane through the three points $P=(1,0,2)$, $Q=(3,-1,3)$, and $R=(1,2,5)$. Show how you obtain a normal vector and justify the resulting equation.

What is the distance between $P=(2,-1,4)$ and $Q=(5,3,4)$?

Compute the dot product of $\mathbf a=\langle 2,-1,4\rangle$ and $\mathbf b=\langle -1,3,1\rangle$.

True or false: If $c<0$, then $c\mathbf v$ points in the opposite direction from the nonzero vector $\mathbf v$.

Which is the unit vector in the direction of $\mathbf v=\langle 3,-4\rangle$?

For $\mathbf u=\langle 2,-1,3\rangle$ and $\mathbf v=\langle 1,2,-2\rangle$, enter the exact value of $\mathbf u\cdot\mathbf v$.

Let $\mathbf u=\langle 1,2\rangle$ and $\mathbf v=\langle 4,1\rangle$. What is the vector projection of $\mathbf v$ onto $\mathbf u$?

Let $\mathbf u=\langle 2,0\rangle$ and $\mathbf v=\langle 3,4\rangle$. Enter the exact scalar projection of $\mathbf v$ onto $\mathbf u$.

Which vector equation represents the line through $P=(2,-1,4)$ and $Q=(0,2,5)$?

Which scalar equation describes the plane through $P_0=(2,1,-1)$ with normal vector $\mathbf n=\langle 1,-2,3\rangle$?

Let $\mathbf u=\langle 1,2,0\rangle$, $\mathbf v=\langle 2,4,0\rangle$, and $\mathbf w=\langle 0,1,1\rangle$. Select all statements that are true.