Past exam of the mathematics course of the University of Oxford / 2021 / Prelims / Mathematics IV resit / 1 / a / ii

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- Solution ii

Solution

Use the convention

[a, b, c] = a \cdot (b \times c)

for the scalar triple product. With

q, w

as above,

q = n \times w

. The vector triple product identity gives

q \times w = - w \times (n \times w) = - n ∣ w ∣^{2} + w (w \cdot n) = - n ∣ w ∣^{2} .

The Lagrange identity for the cross product,

∣ a \times b ∣^{2} = ∣ a ∣^{2} ∣ b ∣^{2} - (a \cdot b)^{2}

, also gives

∣ q ∣^{2} = ∣ v ∣^{2} - (n \cdot v)^{2} = ∣ w ∣^{2} .

Taking the dot product with the unit vector

n

proves

[n, n \times v, n \times (v \times n)] = - ∣ n \times v ∣^{2} .

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