Question

If vectors u1 and u2 are orthogonal, what is the value of ||u1 + u2||²?

A) ||u1||² + ||u2||²
B) 2 ||u1|| ||u2||
C) ||u1||² - ||u2||²
D) ||u1||² + 2 ||u1|| ||u2|| + ||u2||²

Answer 1

The value of ||u1 + u2||² when u1 and u2 are orthogonal vectors is ||u1||² + ||u2||²

Step-by-step explanation:

When vectors u1 and u2 are orthogonal, their dot product is equal to zero. We can use this property to find the value of ||u1 + u2||². Let's represent the dot product of u1 and u2 as u1 · u2. So, u1 · u2 = 0.

To find ||u1 + u2||², we can use the formula: ||u||² = u · u. Substituting the given vectors, we have: ||u1 + u2||² = (u1 + u2) · (u1 + u2).

Expanding the equation, we have: ||u1 + u2||² = u1 · u1 + u1 · u2 + u2 · u1 + u2 · u2.

Since u1 · u2 = 0 and u2 · u1 = 0, we can simplify the equation to: ||u1 + u2||² = u1 · u1 + u2 · u2.

This is equivalent to ||u1||² + ||u2||². Therefore, the value of ||u1 + u2||² is option A) ||u1||² + ||u2||².