Dot product Knowledge-Based Download PDF

Questions

1. 1.

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

   • A) $60^\circ$
   • B) $90^\circ$
   • C) $150^\circ$
   • D) $120^\circ$
2. 2.

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

   • A) $30^\circ$
   • B) $45^\circ$
   • C) $60^\circ$
   • D) $90^\circ$
3. 3.

   Two vectors $\vec{A}$ and $\vec{B}$ have magnitudes of 3 and 4, respectively, and the angle between them is $60^\circ$. What is the value of the scalar product $\vec{A} \cdot (2\vec{A} + 3\vec{B})$?

   • A) 24
   • B) 27
   • C) 54
   • D) 36
4. 4.

   Two non-zero 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) 90°
   • B) 0°
   • C) 45°
   • D) 180°
5. 5.

   Given the vectors $\vec{A} = 3\hat{i} + 2\hat{j}$ and $\vec{B} = -\hat{i} + 4\hat{j}$, what is the value of the scalar product $(\vec{A} + \vec{B}) \cdot (\vec{A} - \vec{B})$?

   • A) 20
   • B) 5
   • C) 4
   • D) -4
6. 6.

   Two vectors $\vec{A}$ and $\vec{B}$ have magnitudes $|\vec{A}| = 5$ and $|\vec{B}| = 5$, and their scalar product is $\vec{A} \cdot \vec{B} = 7$. What is the magnitude of the vector difference $\vec{A} - \vec{B}$?

   • A) 6
   • B) 8
   • C) 10
   • D) 0
7. 7.

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

   • A) 20
   • B) 8
   • C) 5
   • D) 4
8. 8.

   Two vectors are given by $\vec{A} = 3\hat{i} - 4\hat{j}$ and $\vec{B} = 8\hat{i} + c\hat{j}$. If $\vec{A}$ and $\vec{B}$ are perpendicular, what is the value of the scalar constant c?

   • A) -6
   • B) -4
   • C) 4
   • D) 6
9. 9.

   Two vectors $\vec{A}$ and $\vec{B}$ have magnitudes of $4$ and $6$, respectively. If their scalar product $\vec{A} \cdot \vec{B}$ is equal to $-12$, what is the angle between the two vectors?

   • A) $30^\circ$
   • B) $60^\circ$
   • C) $150^\circ$
   • D) $120^\circ$
10. 10.

    Two vectors are given in unit-vector notation as $\vec{u} = 4\hat{i} - 3\hat{j} + 2\hat{k}$ and $\vec{v} = 2\hat{i} + 5\hat{j} - \hat{k}$. What is the scalar product $\vec{u} \cdot \vec{v}$?

    • A) $-9$
    • B) $-5$
    • C) $21$
    • D) $8\hat{i} - 15\hat{j} - 2\hat{k}$