Question

Evaluate the following expression: \[ \mathbf{i} \cdot \mathbf{i} + \mathbf{i} \cdot \mathbf{j} + \mathbf{j} \cdot \mathbf{j} + \mathbf{j} \cdot \mathbf{k} + \mathbf{k} \cdot \mathbf{k} \]

• A. 5
• B. 4
• C. 3
• D. 2

Hint:

Remember that the dot product of a unit vector with itself is 1, and the dot product between two perpendicular unit vectors is 0.

Answer 1

The Correct Option is C

We are asked to evaluate the following expression involving dot products of unit vectors: \[ \mathbf{i} \cdot \mathbf{i} + \mathbf{i} \cdot \mathbf{j} + \mathbf{j} \cdot \mathbf{j} + \mathbf{j} \cdot \mathbf{k} + \mathbf{k} \cdot \mathbf{k} \] Step 1: Recall the properties of dot products for unit vectors: - \(\mathbf{i} \cdot \mathbf{i} = 1\) - \(\mathbf{j} \cdot \mathbf{j} = 1\) - \(\mathbf{k} \cdot \mathbf{k} = 1\) - \(\mathbf{i} \cdot \mathbf{j} = 0\), \(\mathbf{i} \cdot \mathbf{k} = 0\), and \(\mathbf{j} \cdot \mathbf{k} = 0\) because the unit vectors are perpendicular to each other. Step 2: Substitute these values into the expression: \[ 1 + 0 + 1 + 0 + 1 = 3 \] Thus, the correct answer is: (C) 3.