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NSWMaths Extension 1Quick questions

Vectors (ME-V1)

Quick questions on Vector projection: scalar projection a⋅b∣b∣\frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{b}|} and vector projection a⋅b∣b∣2b\frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{b}|^2} \mathbf{b}

13short 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 scalar projection?
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The scalar projection of a\mathbf{a} onto b\mathbf{b} is the signed length of the projection. It is
What is vector projection?
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The vector projection of a\mathbf{a} onto b\mathbf{b} is the actual vector "shadow":
What is geometric picture?
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Imagine a\mathbf{a} and b\mathbf{b} both starting at the origin. Drop a perpendicular from the head of a\mathbf{a} to the line containing b\mathbf{b}. The foot of the perpendicular is at the head of the vector projection of a\mathbf{a} onto b\mathbf{b}.
What is decomposition?
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Any vector a\mathbf{a} can be split into a component parallel to b\mathbf{b} and a component perpendicular to b\mathbf{b}:
What is useful identities?
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If a\mathbf{a} is already parallel to b\mathbf{b}, then projba=a\text{proj}_{\mathbf{b}} \mathbf{a} = \mathbf{a} and the perpendicular part is zero.
What is projection onto a diagonal?
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Find the vector projection of a=(4,2)\mathbf{a} = (4, 2) onto b=(1,1)\mathbf{b} = (1, 1).
What is component perpendicular?
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Find the component of a=(4,2)\mathbf{a} = (4, 2) perpendicular to b=(1,1)\mathbf{b} = (1, 1).
What is force decomposition?
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A force F=(10,5)\mathbf{F} = (10, 5) N acts on an object. The object can only move in the direction d=(3,4)\mathbf{d} = (3, 4). Find the effective force in that direction.
What is scalar projection negative?
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Find the scalar projection of a=(−1,2)\mathbf{a} = (-1, 2) onto b=(3,0)\mathbf{b} = (3, 0).
What is ∣b∣|\mathbf{b}| versus ∣b∣2|\mathbf{b}|^2?
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Scalar projection has ∣b∣|\mathbf{b}| in the denominator; vector projection has ∣b∣2|\mathbf{b}|^2.
What is sign of the scalar projection?
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It can be negative, indicating the projection is in the opposite direction to b\mathbf{b}. The magnitude is the (unsigned) length.
What is projecting onto a unit vector?
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If b\mathbf{b} has ∣b∣=1|\mathbf{b}| = 1, the scalar projection simplifies to a⋅b\mathbf{a} \cdot \mathbf{b} and the vector projection to (a⋅b)b(\mathbf{a} \cdot \mathbf{b}) \mathbf{b}.
What is confusing a⋅b\mathbf{a} \cdot \mathbf{b} with the scalar projection?
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a⋅b\mathbf{a} \cdot \mathbf{b} is the dot product, related to but not equal to the scalar projection unless ∣b∣=1|\mathbf{b}| = 1. :::

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