Vectors and matrices Knowledge-Based Practice

Questions

1. 1.

   A vector $\mathbf{v}$ has a magnitude of 8 and a direction angle of $\frac{5\pi}{6}$ measured counterclockwise from the positive $x$-axis. What is the component form of $\mathbf{v}$?

   • A) $\langle -4\sqrt{3}, 4 \rangle$
   • B) $\langle -4, 4\sqrt{3} \rangle$
   • C) $\langle 4\sqrt{3}, -4 \rangle$
   • D) $\langle 4, -4\sqrt{3} \rangle$
2. 2.

   What is the cosine of the angle between the vectors $\mathbf{u} = \langle 3, 4 \rangle$ and $\mathbf{v} = \langle 5, 12 \rangle$?

   • A) $\frac{33}{65}$
   • B) $\frac{56}{65}$
   • C) $\frac{63}{169}$
   • D) $\frac{63}{65}$
3. 3.

   What is the area of the parallelogram determined by the vectors $\mathbf{u} = \langle 3, -1 \rangle$ and $\mathbf{v} = \langle 2, 4 \rangle$?

   • A) 10
   • B) 12
   • C) 14
   • D) 16
4. 4.

   What is the image of the vector $\mathbf{v} = \begin{pmatrix} 3 \\ -4 \end{pmatrix}$ when transformed by the matrix $A = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$?

   • A) $\begin{pmatrix} -4 \\ -3 \end{pmatrix}$
   • B) $\begin{pmatrix} -3 \\ 4 \end{pmatrix}$
   • C) $\begin{pmatrix} 4 \\ 3 \end{pmatrix}$
   • D) $\begin{pmatrix} 3 \\ 4 \end{pmatrix}$
5. 5.

   What is the vector projection of $\mathbf{u} = \langle 7, 1 \rangle$ onto $\mathbf{v} = \langle 2, 1 \rangle$?

   • A) $\langle 2, 1 \rangle$
   • B) $\langle 4, 2 \rangle$
   • C) $\langle 6, 3 \rangle$
   • D) $\langle 1, -2 \rangle$

Answers and step-by-step solutions appear after you finish the test.