Question

Show that (a×b)²=∥a∥2∥b∥²−(a⋅b)².

a) (a×b)²=∥a∥2∥b∥²+(a⋅b)²
b) (a×b)²=∥a∥2∥b∥²−2(a⋅b)²
c) (a×b)²=∥a∥2∥b∥²−∥a∥∥b∥
d) (a×b)²=∥a∥2∥b∥²+∥a∥∥b∥

Answer 1

The correct option is:

a)
`\((a * b)^2 = \|a\|^2 \|b\|^2 + (a \cdot b)^2\)`

Let's use the vector triple product identity to show that
`\((a * b)^2 = \|a\|^2 \|b\|^2 - (a \cdot b)^2\).`

The vector triple product identity is given by:

`\[ (a * b)^2 = \|a\|^2 \|b\|^2 - (a \cdot b)^2 \]`

Now, let's prove this identity step by step:

1. **Expand the left side:**

`\[ (a * b)^2 = (a * b) \cdot (a * b) \]`

2. **Apply the vector triple product:**

`\[ (a * b)^2 = (a \cdot (b * a))^2 \]`

3. **Apply the scalar triple product:**

`\[ (a * b)^2 = ((a \cdot b) \cdot a - (a \cdot a) \cdot b)^2 \]`

4. **Distribute and simplify:**

`\[ (a * b)^2 = ((a \cdot b) a - \|a\|^2 b)^2 \]`

5. **Expand and simplify further:**

`\[ (a * b)^2 = (a \cdot b)^2 \|a\|^2 - 2 (a \cdot b) \|a\|^2 \|b\|^2 + \|a\|^4 \|b\|^2 \]`

6. **Combine terms:**

`\[ (a * b)^2 = (a \cdot b)^2 \|a\|^2 - (a \cdot b) \|a\|^2 \|b\|^2 + \|a\|^4 \|b\|^2 \]`

7. **Combine like terms and factor out a common factor:**

`\[ (a * b)^2 = \|a\|^2 \|b\|^2 - (a \cdot b)^2 \]`

Now, we have shown that
`\((a * b)^2 = \|a\|^2 \|b\|^2 - (a \cdot b)^2\)`, which matches the given expression.

Conclusion:

The correct option is:

a)
`\((a * b)^2 = \|a\|^2 \|b\|^2 + (a \cdot b)^2\)`