Vector Calculus Questions "Dot Product " & "Cross Product"

on July 24, 2024

Vector Products

Vector Products

Dot Product Questions

1. Given vectors \(\mathbf{A} = 2\mathbf{i} + 3\mathbf{j}\) and \(\mathbf{B} = -\mathbf{i} + 4\mathbf{j}\), calculate \(\mathbf{A} \cdot \mathbf{B}\).

2. Find the angle between the vectors \(\mathbf{C} = 5\mathbf{i} + 2\mathbf{j} + \mathbf{k}\) and \(\mathbf{D} = 3\mathbf{i} - 4\mathbf{j} + 2\mathbf{k}\) using their dot product.

3. Vectors \(\mathbf{E} = 7\mathbf{i} - \mathbf{j} + 2\mathbf{k}\) and \(\mathbf{F} = 4\mathbf{i} + \mathbf{j} - 3\mathbf{k}\) are given. Compute the dot product \(\mathbf{E} \cdot \mathbf{F}\).

4. If \(\mathbf{G} = a\mathbf{i} + 3\mathbf{j} - \mathbf{k}\) and \(\mathbf{H} = 2\mathbf{i} + b\mathbf{j} + 4\mathbf{k}\) are perpendicular vectors, find the values of \(a\) and \(b\).

5. Given the vectors \(\mathbf{I} = 2\mathbf{i} + 2\mathbf{j} + 2\mathbf{k}\) and \(\mathbf{J} = x\mathbf{i} + y\mathbf{j} + z\mathbf{k}\) such that \(\mathbf{I} \cdot \mathbf{J} = 0\) and \(\mathbf{J}\) is parallel to the plane \(2x + 2y + 2z = 6\), find the components \(x\), \(y\), and \(z\).

Cross Product Questions

6. Calculate the cross product of the vectors \(\mathbf{A} = 3\mathbf{i} + 2\mathbf{j} - \mathbf{k}\) and \(\mathbf{B} = \mathbf{i} - 4\mathbf{j} + 2\mathbf{k}\).

7. Given vectors \(\mathbf{C} = \mathbf{i} + \mathbf{j}\) and \(\mathbf{D} = \mathbf{i} - \mathbf{j} + \mathbf{k}\), find \(\mathbf{C} \times \mathbf{D}\).

8. Find the area of the parallelogram formed by the vectors \(\mathbf{E} = 2\mathbf{i} + \mathbf{j} + 3\mathbf{k}\) and \(\mathbf{F} = \mathbf{i} - 2\mathbf{j} + \mathbf{k}\) using their cross product.

9. Vectors \(\mathbf{G} = a\mathbf{i} + b\mathbf{j} + \mathbf{k}\) and \(\mathbf{H} = \mathbf{i} + \mathbf{j} + \mathbf{k}\) are parallel. If their cross product is zero, determine the relationship between \(a\) and \(b\).

10. Given the vectors \(\mathbf{I} = 4\mathbf{i} + 2\mathbf{j} - 3\mathbf{k}\) and \(\mathbf{J} = 3\mathbf{i} - \mathbf{j} + \mathbf{k}\), compute the volume of the parallelepiped formed by the vectors \(\mathbf{I}\), \(\mathbf{J}\), and \(\mathbf{K} = \mathbf{i} + \mathbf{j} + 2\mathbf{k}\).

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