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Consider three vectors →x, →y, and →z, each having a magnitude of √2, with the angle between any two of them being π/3. If →a is a non-zero vector orthogonal to both →y and →z, and →b is a non-zero vector orthogonal to both →x and →z, which of the following is true?

1. →b = (→b ⋅ →z) / (→z − →x)
2. →a = (→a ⋅ →y) / (→y − →z)
3. →a ⋅ →b = (→a ⋅ →y)(→b ⋅ →z)
4. →a = −(→a ⋅ →y)(→z − →y)

Correct answer: →a ⋅ →b = (→a ⋅ →y)(→b ⋅ →z)

Solution

The vector →a is orthogonal to both →y and →z, and →b is orthogonal to both →x and →z. The scalar product →a ⋅ →b is derived using the orthogonality conditions and simplifies to (→a ⋅ →y)(→b ⋅ →z).

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