Practice Flashcards
How do you compute the scalar (dot) product of v and w?
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All Flashcards in Topic 3.13
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3.13.18 cards
How do you compute the scalar (dot) product of v and w?
Multiply matching components and add: v·w = v₁w₁ + v₂w₂ + v₃w₃. The result is a single number.
Is the dot product a vector or a number?
A number (scalar) — that's why it's called the scalar product.
What is the formula linking the dot product to the angle?
v·w = |v||w| cos θ, so cos θ = (v·w)/(|v||w|).
How do you find the angle between two vectors?
θ = cos⁻¹[ (v·w)/(|v||w|) ] — dot product over the product of the magnitudes.
What does a NEGATIVE dot product tell you about the angle?
The angle is obtuse (between 90° and 180°), because cos θ is negative.
How do you test if two vectors are perpendicular?
They are perpendicular exactly when v·w = 0.
Find (1, 2, −2)·(3, 0, 1).
(1)(3) + (2)(0) + (−2)(1) = 3 + 0 − 2 = 1.
To find a triangle's angle at vertex A, which vectors do you dot?
AB and AC (both pointing OUT from A); then cos A = (AB·AC)/(|AB||AC|).
3.13.28 cards
What does the cross product v×w give?
A VECTOR perpendicular to both v and w (the dot product gives a scalar).
Component formula for v×w?
v×w = (v₂w₃ − v₃w₂, v₃w₁ − v₁w₃, v₁w₂ − v₂w₁).
What does |v×w| equal?
|v×w| = |v||w| sin θ = the area of the parallelogram with sides v and w.
Area of triangle ABC using vectors?
½|AB×AC| — two sides from the same vertex, crossed, length halved.
How can you check a cross product is right?
It must be perpendicular to both inputs: v·(v×w) = 0 and w·(v×w) = 0.
Dot product vs cross product — what comes out?
Dot → a scalar (number); cross → a vector.
Is the cross product commutative?
No — w×v = −(v×w) (opposite direction); and v×v = 0.
When is |v×w| = 0?
When v and w are parallel (sin θ = 0) — they span no area.
Topic 3.13 study notes
Full notes & explanations for Scalar & vector products (HL only)
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