Hint
Both vectors are nonzero. Multiply corresponding components and add.
Worked solution
(a)
\(\displaystyle \cos\theta=\frac{3 \sqrt{102}}{34}\); use the inverse cosine with the directed-vector sign retained.
Answer: \(\displaystyle \operatorname{acos}{\left(\frac{3 \sqrt{102}}{34} \right)}\)
(b)
Take u·u.
\(17\)
Answer: \(17\)
(c)
Square the dot-product angle formula.
\(\displaystyle \frac{27}{34}\)
Answer: \(\displaystyle \frac{27}{34}\)
Maths4U indicative marking guidance
Maths4U indicative marking guidance, not an official mark scheme. Credit complete valid alternatives and consequential reasoning; an unexplained final answer does not earn a method mark.
(a) · 2 marks
- 1 mark: Compute the scalar product and both nonzero vector magnitudes, then use their signed cosine ratio.
- 1 mark: Give all results requested in this part, with the correct signs, exact values, units, branch and boundary inclusion where applicable.
Alternative methods: Any complete mathematically valid alternative method with the same restrictions and equivalent exact result is acceptable.
Common error: The dot product is a scalar, not a vector.
(b) · 2 marks
- 1 mark: Take u·u.
- 1 mark: Give the correct requested result, retaining signs, units and any domain restrictions.
Alternative methods: Any complete mathematically valid alternative method with the same restrictions and equivalent exact result is acceptable.
Common error: A negative dot product represents an obtuse directed angle.
(c) · 2 marks
- 1 mark: Square the dot-product angle formula.
- 1 mark: Give the correct requested result, retaining signs, units and any domain restrictions.
Alternative methods: Any complete mathematically valid alternative method with the same restrictions and equivalent exact result is acceptable.
Common error: The dot product is a scalar, not a vector.
Total: 6 marks.