Practice Flashcards
Dot product of v = (v₁,v₂,v₃) and w = (w₁,w₂,w₃) in components?
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All Flashcards in Topic 3.13
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3.13.18 cards
Dot product of v = (v₁,v₂,v₃) and w = (w₁,w₂,w₃) in components?
v·w = v₁w₁ + v₂w₂ + v₃w₃ — multiply matching components and add. The answer is a number.
Geometric formula for the dot product?
v·w = |v||w|cos θ, where θ is the angle between the vectors.
How do you find the angle between two vectors?
cos θ = (v·w)/(|v||w|), then θ = cos⁻¹ of that value.
Is the dot product a vector or a number?
A number (a scalar) — that's why it's called the SCALAR product.
Magnitude of a vector (v₁,v₂,v₃)?
|v| = √(v₁² + v₂² + v₃²) — the length of the arrow.
a = (2,−1,3), b = (4,0,−2): find a·b.
(2)(4)+(−1)(0)+(3)(−2) = 8 + 0 − 6 = 2.
If the dot product of two vectors is 0, what is the angle?
90° — they are perpendicular.
u = (1,2,2), v = (2,0,−1): find the angle between them.
u·v = 0, so cos θ = 0 and θ = 90°.
3.13.28 cards
When are two vectors perpendicular?
When their dot product is 0 (because v·w = |v||w|cos 90° = 0).
When are two vectors parallel?
When one is a scalar multiple of the other: v = t w (components in the same ratio).
a = (3, k, 2), b = (1, −4, 5) are perpendicular. Find k.
a·b = 3 − 4k + 10 = 0 ⇒ k = 13/4.
Test: are (6, −9) and (2, −3) parallel?
Yes — (6, −9) = 3(2, −3), a scalar multiple.
What angle do parallel vectors make? Perpendicular?
Parallel: 0° (same way) or 180° (opposite). Perpendicular: 90°.
A vector perpendicular to (3, 4) in 2-D?
Swap and negate one entry: (−4, 3) (or (4, −3)); check (3)(−4)+(4)(3)=0.
How do you find an unknown component for perpendicular vectors?
Set the dot product equal to 0 and solve the resulting equation for the unknown.
If u = t v, what does that tell you about u and v?
They are parallel (u is a scaled copy of v).
Topic 3.13 study notes
Full notes & explanations for Scalar product (HL only)
Math AA exam skills
Paper structures, command terms & tips
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