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NSWMaths Extension 1Quick questions

Vectors (ME-V1)

Quick questions on Vector arithmetic: addition, scalar multiplication, magnitude and unit vectors

14short 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 notation?
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A two-dimensional vector a\mathbf{a} has horizontal component a1a_1 and vertical component a2a_2. Common notations:
What is vector addition and subtraction?
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$a+b=(a1+b1a2+b2),a−b=(a1−b1a2−b2).\mathbf{a} + \mathbf{b} = \begin{pmatrix} a_1 + b_1 \\ a_2 + b_2 \end{pmatrix}, \qquad \mathbf{a} - \mathbf{b} = \begin{pmatrix} a_1 - b_1 \\ a_2 - b_2 \end{pmatrix}.$
What is scalar multiplication?
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For a scalar λ\lambda and vector a\mathbf{a},
What is magnitude?
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The magnitude (length) of a\mathbf{a} is
What is unit vector?
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The unit vector in the direction of a\mathbf{a} (assumed non-zero) is
What is vector laws?
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Vector addition is commutative (a+b=b+a\mathbf{a} + \mathbf{b} = \mathbf{b} + \mathbf{a}) and associative ((a+b)+c=a+(b+c)(\mathbf{a} + \mathbf{b}) + \mathbf{c} = \mathbf{a} + (\mathbf{b} + \mathbf{c})).
What is direction vector and unit vector?
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Let v=(6,8)\mathbf{v} = (6, 8). Find the unit vector in the direction of v\mathbf{v}.
What is vector from two points?
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Find the vector from A=(1,2)A = (1, 2) to B=(4,6)B = (4, 6) and its magnitude.
What is scalar combination?
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If a=(2,3)\mathbf{a} = (2, 3) and b=(5,−1)\mathbf{b} = (5, -1), find 3a−2b3 \mathbf{a} - 2 \mathbf{b}.
What is parallel vectors?
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Are a=(2,−1)\mathbf{a} = (2, -1) and b=(−6,3)\mathbf{b} = (-6, 3) parallel?
What is forgetting to take the square root for magnitude?
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∣a∣=a12+a22|\mathbf{a}| = \sqrt{a_1^2 + a_2^2}, not a12+a22a_1^2 + a_2^2.
What is mixing notations?
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(a1,a2)(a_1, a_2), (a1a2)\begin{pmatrix} a_1 \\ a_2 \end{pmatrix}, a1i+a2ja_1 \mathbf{i} + a_2 \mathbf{j} all denote the same vector. Use the convention asked or default to column form.
What is unit vector confusion?
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A unit vector has magnitude 11, not magnitude equal to one of the components.
What is adding magnitudes?
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∣a+b∣≠∣a∣+∣b∣|\mathbf{a} + \mathbf{b}| \neq |\mathbf{a}| + |\mathbf{b}| in general; the triangle inequality only gives ∣a+b∣≤∣a∣+∣b∣|\mathbf{a} + \mathbf{b}| \le |\mathbf{a}| + |\mathbf{b}|. :::

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