Dot product Knowledge-Based Practice

Questions

1. 1.

   Given three vectors such that $\vec{A} \cdot \vec{B} = 7$ and $\vec{A} \cdot \vec{C} = -4$, what is the value of $\vec{A} \cdot (3\vec{B} + 2\vec{C})$?

   • A) 29
   • B) 13
   • C) 3
   • D) -13
2. 2.

   The scalar product of two vectors $\vec{A}$ and $\vec{B}$ is $\vec{A} \cdot \vec{B} = 15$. What is the value of the scalar product $(3\vec{A}) \cdot (-2\vec{B})$?

   • A) 90
   • B) -45
   • C) -90
   • D) 15
3. 3.

   What is the angle between the vectors $\vec{A} = \hat{i} + \hat{j} + 2\hat{k}$ and $\vec{B} = -\hat{i} + 2\hat{j} + \hat{k}$?

   • A) 30°
   • B) 45°
   • C) 60°
   • D) 90°
4. 4.

   What is the vector projection of $\vec{A} = 4\hat{i} + 7\hat{j}$ onto $\vec{B} = 2\hat{i} + \hat{j}$?

   • A) $6\hat{i} + 3\hat{j}$
   • B) $-2\hat{i} + 4\hat{j}$
   • C) $8\hat{i} + 7\hat{j}$
   • D) $3\hat{i} + 6\hat{j}$
5. 5.

   Two vectors $\vec{A}$ and $\vec{B}$ have magnitudes of 5 and 12, respectively. What is the minimum possible value of their scalar product $\vec{A} \cdot \vec{B}$?

   • A) 0
   • B) -60
   • C) 7
   • D) -1
6. 6.

   A vector $\vec{A}$ lies in the xy-plane and has a magnitude of 10. If $\vec{A} \cdot \hat{i} = 8$ and its y-component is positive, what is the vector $\vec{A}$ in unit-vector notation?

   • A) $8\hat{i} + 6\hat{j}$
   • B) $6\hat{i} + 8\hat{j}$
   • C) $8\hat{i} + 2\hat{j}$
   • D) $8\hat{i} - 6\hat{j}$
7. 7.

   Two nonzero vectors $\vec{A}$ and $\vec{B}$ satisfy the condition $|\vec{A} + \vec{B}| = |\vec{A} - \vec{B}|$. What is the angle between $\vec{A}$ and $\vec{B}$?

   • A) $0^\circ$
   • B) $45^\circ$
   • C) $90^\circ$
   • D) $180^\circ$
8. 8.

   What is the angle between the vector $\vec{A} = \hat{i} + \hat{j} + \sqrt{2}\hat{k}$ and the positive z-axis?

   • A) $45^\circ$
   • B) $30^\circ$
   • C) $60^\circ$
   • D) $90^\circ$
9. 9.

   Vector $\vec{A}$ has a magnitude of 10 and is directed at $15^\circ$ above the positive x-axis, while vector $\vec{B}$ has a magnitude of 6 and is directed at $75^\circ$ above the positive x-axis. What is the dot product $\vec{A} \cdot \vec{B}$?

   • A) 0
   • B) $30\sqrt{3}$
   • C) 60
   • D) 30
10. 10.

    Two vectors $\vec{A}$ and $\vec{B}$ have magnitudes of 5 and 4, respectively. If the magnitude of their vector difference is $|\vec{A} - \vec{B}| = 3$, what is the value of the dot product $\vec{A} \cdot \vec{B}$?

    • A) 8
    • B) 12
    • C) 16
    • D) 20
11. 11.

    Given two vectors $\vec{a}$ and $\vec{b}$ with magnitudes $|\vec{a}| = 3$ and $|\vec{b}| = 4$, and scalar product $\vec{a} \cdot \vec{b} = -6$, what is the value of the product $(\vec{a} + 2\vec{b}) \cdot (2\vec{a} - \vec{b})$?

    • A) -32
    • B) -14
    • C) 4
    • D) 32
12. 12.

    What is the scalar projection of the vector $\vec{A} = 6\hat{i} + 2\hat{j}$ onto the vector $\vec{B} = 4\hat{i} + 3\hat{j}$?

    • A) 6
    • B) 30
    • C) 5
    • D) 10
13. 13.

    For what value of the scalar constant $c$ are the vectors $\vec{A} = 3\hat{i} - 6\hat{j}$ and $\vec{B} = 4\hat{i} + c\hat{j}$ perpendicular to each other?

    • A) $-2$
    • B) 0
    • C) 2
    • D) 6
14. 14.

    Two vectors, $\vec{u}$ and $\vec{v}$, have magnitudes of 6 and 5, respectively. If their dot product $\vec{u} \cdot \vec{v} = -15$, what is the angle between the two vectors?

    • A) 60°
    • B) 120°
    • C) 150°
    • D) 30°
15. 15.

    Given the two vectors $\vec{A} = 4\hat{i} - 2\hat{j} + 3\hat{k}$ and $\vec{B} = -2\hat{i} + 5\hat{j} + \hat{k}$, what is the value of their dot product $\vec{A} \cdot \vec{B}$?

    • A) 5
    • B) 15
    • C) -21
    • D) -15

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