5B (Vectors and Matrices)

For

A = (A_{1}, A_{2}, A_{3})

and

B = (B_{1}, B_{2}, B_{3})

, the scalar product is the inner product

A \cdot B = A_{1} B_{1} + A_{2} B_{2} + A_{3} B_{3} .

(1)

The cross product is

A \times B = (A_{2} B_{3} - A_{3} B_{2},, A_{3} B_{1} - A_{1} B_{3},, A_{1} B_{2} - A_{2} B_{1}) .

(2)

It is perpendicular to both vectors and has magnitude

∣ A ∣∣ B ∣ sin θ

ii

 0  0

Solution

 0  0

The vector triple product identity gives

A \times (B \times C) = B (A \cdot C) - C (A \cdot B) .

(1)

Applying the same identity after reversing the outer cross product gives

(A \times B) \times C = B (A \cdot C) - A (B \cdot C) .

(2)

The formulas differ because the cross product is not associative.

iii

 0  0

Solution

 0  0

Let

Δ = A \cdot (B \times C) .

(1)

Linear independence implies that the scalar triple product

De lt a

is nonzero. Applying the vector triple-product formulas to

(A \times B) \times (C \times D)

recovers the coefficients of

D

in the basis

A, B, C

. Equivalently, the reciprocal basis to

A, B, C

\frac{B \times C}{Δ}, \frac{C \times A}{Δ}, \frac{A \times B}{Δ} .

(2)

Taking scalar products with

D

therefore yields

D = \frac{D \cdot ( B \times C )}{Δ} A + \frac{D \cdot ( C \times A )}{Δ} B + \frac{D \cdot ( A \times B )}{Δ} C .

(3)

b

 0  0

Solution

 0  0

Because the sphere passes through the origin and has centre

P

, its radius is

∣ P ∣

. A point

X

lies on it exactly when

∣ X - P ∣^{2} = ∣ P ∣^{2},

(1)

or equivalently

2 P \cdot X = ∣ X ∣^{2} .

(2)

With

P = α A + βB + γ C

, the three required equations are the Gram matrix system

A \cdot A B \cdot A C \cdot A A \cdot B B \cdot B C \cdot B A \cdot C B \cdot C C \cdot C α β γ = \frac{1}{2} ∣ A ∣^{2} ∣ B ∣^{2} ∣ C ∣^{2} .

(3)

For the specified vectors this becomes

α + β = \frac{1}{2}, α + 2 β + γ = 1, β + 5 γ = \frac{5}{2} .

(4)

Thus

α = \frac{1}{2}, β = 0, γ = \frac{1}{2},

(5)

and the centre is

P = \frac{1}{2} A + \frac{1}{2} C = (\frac{1}{2}, \frac{1}{2}, 1) .

(6)

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Past exam of the mathematics course of the University of Cambridge / 2022 / ia / Paper 1 / 5B (Vectors and Matrices)

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