Cross product Knowledge-Based Download PDF

Questions

1. 1.

   Two vectors $\vec{A} = 4\hat{i}$ and $\vec{B} = 2\hat{i} + 5\hat{j}$ define adjacent sides of a parallelogram. What is the area of this parallelogram?

   • A) 20 square units
   • B) 10 square units
   • C) 8 square units
   • D) 28 square units
2. 2.

   For two non-zero vectors $\vec{A}$ and $\vec{B}$, the magnitude of their cross product is equal to their scalar product such that $|\vec{A} \times \vec{B}| = \vec{A} \cdot \vec{B}$. What is the angle between the two vectors?

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

   Which of the following Cartesian unit vector cross products results in $-\hat{i}$?

   • A) $\hat{j} \times \hat{k}$
   • B) $\hat{i} \times \hat{k}$
   • C) $\hat{j} \times \hat{i}$
   • D) $\hat{k} \times \hat{j}$
4. 4.

   Which of the following statements is always true regarding the cross product of two non-zero, non-collinear vectors $\vec{A}$ and $\vec{B}$?

   • A) The resulting vector lies within the plane formed by $\vec{A}$ and $\vec{B}$.
   • B) The resulting vector is perpendicular to both $\vec{A}$ and $\vec{B}$.
   • C) Reversing the order of vectors preserves the result such that $\vec{A} \times \vec{B} = \vec{B} \times \vec{A}$.
   • D) The magnitude of the cross product reaches its maximum value when $\vec{A}$ and $\vec{B}$ are parallel.
5. 5.

   Two vectors $\vec{A}$ and $\vec{B}$ have magnitudes of 8 units and 5 units, respectively. If the angle between them is 30°, what is the magnitude of their cross product, $|\vec{A} \times \vec{B}|$?

   • A) 20 units
   • B) $20\sqrt{3}$ units
   • C) 40 units
   • D) 10 units
6. 6.

   Given the vectors $\vec{u} = 2\hat{i} - \hat{j} + 3\hat{k}$ and $\vec{v} = \hat{i} + 2\hat{j} - \hat{k}$, what is the cross product $\vec{u} \times \vec{v}$?

   • A) $5\hat{i} - 5\hat{j} - 5\hat{k}$
   • B) $-5\hat{i} - 5\hat{j} + 5\hat{k}$
   • C) $-5\hat{i} + 5\hat{j} + 5\hat{k}$
   • D) $7\hat{i} + \hat{j} + 3\hat{k}$