A dot product of zero means a right angle: The dot product is a built-in right-angle detector: v·w = 0 ⟺ v and w are perpendicular (also written v ⊥ w).
Why does v·w = 0 mean perpendicular?
Start from the angle form
v·w = |v||w| cos θ.
The lengths aren't zero
So the only way the product can be zero is cos θ = 0.
So the angle is 90°
cos θ = 0 means θ = 90°.
IB-style question — find an unknown for perpendicularity
Vectors are a = (3, k, 2) and b = (1, −4, 5).
Given that a is perpendicular to b, find the value of k.
Step by step
- Perpendicular means the dot product is zero.
- Write the dot product with k inside.
- Simplify the constant terms.
- Solve for k.
Final answer
k = 13/4 (then a·b = 0, so a ⊥ b).
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Parallel vectors point the same way — one is a stretch of the other: Two non-zero vectors are parallel when one is a scalar multiple of the other: v = t w for some number t.
What parallel means
- The components are then proportional — each entry of v is the same multiple t of the matching entry of w.
- Parallel means θ = 0° or 180°, so cos θ = ±1.
- Contrast with perpendicular: this is the opposite extreme to the dot-product-zero case.
IB-style question — find an unknown for parallel
Vectors are u = (4, 6, −2) and v = (2, m, −1).
Given that u is parallel to v, find the value of m.
Step by step
- Parallel means u = t·v for some scalar t. Compare the first components.
- Check with the third components (should also give t = 2).
- Use t = 2 on the middle components.
- Solve for m.
Final answer
m = 3 (then u = 2v, so they are parallel).
IB-style question — perpendicular vector of given length
Find a vector q that is perpendicular to p = (3, 4) and has magnitude 10.
Step by step
- Swap and negate one component to get a perpendicular direction (check: (3)(−4)+(4)(3)=0).
- Its magnitude is √((−4)²+3²) = 5, so scale by 10/5 = 2 to reach length 10.
- Check the magnitude.
Final answer
q = (−8, 6) (or its opposite (8, −6)).