Question

a ⃗,b ⃗,c ⃗are non-zero vectors. If a ⃗×b ⃗= a ⃗×c ⃗ and a ⃗ .b ⃗= a ⃗ .c ⃗ then show that b ⃗= c ⃗.

Answer 1

Answer on Question #44084 – Math - Vector Calculus

a ¬, b ¬, c ¬ are non-zero vectors. If a ¬×b ¬ = a ¬×c ¬ and a ¬.b ¬ = a ¬.c ¬ then show that b ¬ = c ¬.

Solution

$$\vec{a} \times (\vec{b} \times \vec{c}) = \vec{b}(\vec{a} \cdot \vec{c}) - \vec{c}(\vec{a} \cdot \vec{b}).$$ $$\vec{a} \times (\vec{b} - \vec{c}) = \vec{a} \times \vec{b} - \vec{a} \times \vec{c} = \vec{0}.$$ $$\vec{a} \times (\vec{a} \times (\vec{b} - \vec{c})) = \vec{a} \times \vec{0} = \vec{0}.$$

But

$$\vec{a} \times (\vec{a} \times (\vec{b} - \vec{c})) = \vec{a}(\vec{a} \cdot (\vec{b} - \vec{c})) - (\vec{b} - \vec{c})(\vec{a} \cdot \vec{a}) = \vec{a}(\vec{a} \cdot \vec{b} - \vec{a} \cdot \vec{c}) - (\vec{b} - \vec{c})(\vec{a} \cdot \vec{a}) = \vec{a}(0) - (\vec{b} - \vec{c})(\vec{a} \cdot \vec{a}) = -(\vec{b} - \vec{c})(\vec{a} \cdot \vec{a}) = \vec{0}.$$

$\vec{a}$ is non-zero vector, so

$$(\vec{a} \cdot \vec{a}) \neq 0.$$

That’s why

$$(\vec{b} - \vec{c}) = \vec{0} \quad \text{and} \quad \vec{b} = \vec{c}.$$

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