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NSWMaths Extension 1Quick questions

Vectors (ME-V1)

Quick questions on The scalar (dot) product: component formula, geometric formula, angle between vectors and orthogonality

11short 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 two equivalent formulas?
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For two-dimensional vectors a=(a1,a2)\mathbf{a} = (a_1, a_2) and b=(b1,b2)\mathbf{b} = (b_1, b_2), the scalar product (dot product) is
What is the angle between two vectors?
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Solving the geometric formula for cos⁡θ\cos \theta,
What is orthogonality?
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Two non-zero vectors are perpendicular (orthogonal) if and only if a⋅b=0\mathbf{a} \cdot \mathbf{b} = 0.
What is properties?
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The scalar product is commutative (a⋅b=b⋅a\mathbf{a} \cdot \mathbf{b} = \mathbf{b} \cdot \mathbf{a}), bilinear (linear in each argument), and gives a⋅a=∣a∣2\mathbf{a} \cdot \mathbf{a} = |\mathbf{a}|^2.
What is the geometric meaning?
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a⋅b\mathbf{a} \cdot \mathbf{b} measures the extent to which a\mathbf{a} and b\mathbf{b} point in the same direction.
What is angle in degrees?
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Find the angle between (2,0)(2, 0) and (1,1)(1, 1).
What is magnitude from a dot product?
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If ∣a∣=3|\mathbf{a}| = 3, ∣b∣=4|\mathbf{b}| = 4, and a⋅b=6\mathbf{a} \cdot \mathbf{b} = 6, find ∣a+b∣|\mathbf{a} + \mathbf{b}|.
What is squaring without expanding?
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∣a+b∣2=∣a∣2+2a⋅b+∣b∣2|\mathbf{a} + \mathbf{b}|^2 = |\mathbf{a}|^2 + 2 \mathbf{a} \cdot \mathbf{b} + |\mathbf{b}|^2, just like (a+b)2=a2+2ab+b2(a + b)^2 = a^2 + 2 a b + b^2.
What is forgetting bilinearity?
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Manipulating a⋅(b+c)\mathbf{a} \cdot (\mathbf{b} + \mathbf{c}) correctly requires the distributive law: a⋅b+a⋅c\mathbf{a} \cdot \mathbf{b} + \mathbf{a} \cdot \mathbf{c}.
What is angle outside [0,π][0, \pi]?
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The angle between two vectors is conventionally in [0,π][0, \pi]. The arccos⁡\arccos function returns this range.
What is mistaking a dot product for a vector?
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The result of a⋅b\mathbf{a} \cdot \mathbf{b} is a scalar. You cannot "add a vector to a dot product". :::

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