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§-Quick questions
NSWMaths Extension 1Vectors (ME-V1)

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

6short 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 the sign of the dot product tells you the angle?
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Because ∣a∣|\mathbf{a}| and ∣b∣|\mathbf{b}| are positive, the sign of a⋅b\mathbf{a} \cdot \mathbf{b} matches the sign of cos⁡θ\cos\theta. So without computing the angle at all you can read off whether it is acute, right or obtuse.
What are finding the angle between two vectors?
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Rearranging the geometric formula gives the angle directly:
What is acute angle: the dot product is positive?
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The vectors broadly agree in direction, cos⁡θ>0\cos\theta > 0, so a⋅b>0\mathbf{a} \cdot \mathbf{b} > 0.
What is right angle: the dot product is zero?
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Perpendicular vectors have cos⁡90∘=0\cos 90^\circ = 0, so a⋅b=0\mathbf{a} \cdot \mathbf{b} = 0. This is the orthogonality test.
What is obtuse angle: the dot product is negative?
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The vectors broadly oppose, cos⁡θ<0\cos\theta < 0, so a⋅b<0\mathbf{a} \cdot \mathbf{b} < 0.
What is angle outside [0,π][0, \pi]?
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The angle between two vectors is conventionally in [0,180∘][0, 180^\circ], which is exactly the range arccos⁡\arccos returns. A negative cosine gives an obtuse angle, not a reflex one.

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