Question

Find the value of the expression \( i \cdot i + j \cdot j + 2k \cdot k \).

Hint:

When working with unit vectors, remember that the dot product of a unit vector with itself is 1, and the dot product of orthogonal unit vectors is 0.

Answer 1

We know the following properties of the unit vectors \( i, j, k \): \[ i \cdot i = 1, j \cdot j = 1, k \cdot k = 1 \] and \[ i \cdot j = 0, j \cdot k = 0, i \cdot k = 0 \text{(since they are orthogonal)}. \] The given expression is: \[ i \cdot i + j \cdot j + 2k \cdot k \] Substitute the values: \[ = 1 + 1 + 2(1) = 1 + 1 + 2 = 4 \] Thus, \[ \boxed{4} \]