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VICSpecialist MathematicsQuick questions

Unit 3: Vectors

Quick questions on Cross product and applications: VCE Specialist Mathematics Unit 3

7short Q&A pairs drawn directly from our worked dot-point answer. For full context and worked exam questions, read the parent dot-point page.

What is normal vector?
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Since a×b\mathbf{a} \times \mathbf{b} is perpendicular to both, it provides a normal to the plane containing them, which is exactly what is needed to write the plane's equation.
What is area?
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The parallelogram on a\mathbf{a} and b\mathbf{b} has area ∣a×b∣|\mathbf{a} \times \mathbf{b}|, and the triangle with those two sides has area 12∣a×b∣\frac{1}{2}|\mathbf{a} \times \mathbf{b}|.
What is parallel test?
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If a\mathbf{a} and b\mathbf{b} are parallel, θ=0\theta = 0 or π\pi, so sin⁡θ=0\sin\theta = 0 and a×b=0\mathbf{a} \times \mathbf{b} = \mathbf{0}. Conversely a zero cross product (for non-zero vectors) means they are parallel.
What are order matters?
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a×b=−b×a\mathbf{a} \times \mathbf{b} = -\mathbf{b} \times \mathbf{a}, so swapping the vectors reverses the normal's direction.
What is q1?
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Compute (1,0,0)×(0,1,0)(1, 0, 0) \times (0, 1, 0). [1 mark]
What is q2?
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Find the area of the parallelogram on (2,0,0)(2, 0, 0) and (0,3,0)(0, 3, 0). [2 marks]
What is q3?
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Show (2,4,6)(2, 4, 6) and (1,2,3)(1, 2, 3) are parallel using the cross product. [2 marks]

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