For any vector r, what is (r·î)(r×î) + (r·ĵ)(r×ĵ) + (r·k̂)(r×k̂) equal to?

Let r = xî + yĵ + zk̂. Then r·î = x, r·ĵ = y, r·k̂ = z. Also r×î = zĵ – yk̂ (wait: r×î = (xî + yĵ + zk̂)×î = y(ĵ×î) + z(k̂×î) = –yk̂ + zĵ). Similarly compute each term and sum: x(zĵ – yk̂) + y(r×ĵ) + z(r×k̂). Working systematically: r×î = zĵ – yk̂; r×ĵ = –zî + xk̂; r×k̂ = yî – xĵ. Then x(zĵ – yk̂) + y(–zî + xk̂) + z(yî – xĵ) = xzĵ – xyk̂ – yzî + xyk̂ + yzî – xzĵ = 0. So the answer is 0.