Vectors in the Plane: Precalculus Math Test Answer Key – PDF Download

Precalculus Vectors Test Answer Key: questions 1–5

Question 1: Components and magnitude (2 points)

1. \(\langle 6, -8\rangle\) (\(4-(-2)=6\), \(-5-3=-8\))
2. \(\sqrt{36+64}=10\)

Scoring: 1 point for each correct part; subtracting the coordinates in the wrong order earns 0 for part (a).

Question 2: Magnitude and direction (2 points)

1. \(\sqrt{9+27}=\sqrt{36}=6\)
2. \(\tan\theta=\frac{3\sqrt{3}}{-3}=-\sqrt{3}\), reference angle \(60^{\circ}\), quadrant II: \(\theta=120^{\circ}\)

Scoring: 1 point for each part; for part (b) award the point only if the angle is in the correct quadrant (60° earns 0).

Question 3: Combining vectors (2 points)

1. \(\langle 6-8,\ -15+6\rangle=\langle -2, -9\rangle\)
2. \(\langle 2+8,\ -5-6\rangle=\langle 10, -11\rangle\)

Scoring: 1 point for each correct vector; 0 for a part with a sign error in the subtraction.

Question 4: Parallelogram rule (3 points)

1. \(\vec{a}=\langle 3, 2\rangle\), \(\vec{b}=\langle -1, 3\rangle\), so \(\vec{a}+\vec{b}=\langle 2, 5\rangle\)
2. \(\vec{a}-\vec{b}=\langle 4, -1\rangle\)
3. \(\sqrt{4+25}=\sqrt{29}\)

Scoring: 1 point per part. In part (a) the point requires correct components for both vectors and the sum.

Question 5: Unit vector (2 points)

1. \(\sqrt{81+144}=15\)
2. \(\frac{1}{15}\langle -9, 12\rangle=\langle -\frac{3}{5}, \frac{4}{5}\rangle\)

Scoring: 1 point per part; part (b) requires dividing every component by the magnitude.

Precalculus Vectors Test Answer Key: questions 6–9

Question 6: Dot product and angle (3 points)

1. \(2(-1)+4(3)=10\)
2. \(|\vec{u}|=\sqrt{20}=2\sqrt{5}\) and \(|\vec{v}|=\sqrt{10}\)
3. \(\cos\theta=\frac{10}{\sqrt{20}\sqrt{10}}=\frac{10}{\sqrt{200}}=\frac{\sqrt{2}}{2}\), so \(\theta=45^{\circ}\)

Scoring: 1 point per part.

Question 7: Perpendicular vectors (2 points)

Perpendicular means dot product 0: \(10k-20=0\) (1 point), so \(k=2\). Check: \(\langle 2, 5\rangle\cdot\langle 10, -4\rangle=20-20=0\).

Scoring: 1 point for the equation, 1 point for the value and the check.

Question 8: Airplane and wind (2 points)

Ground velocity \(\langle 70, 240\rangle\), speed \(\sqrt{4900+57600}=\sqrt{62500}=\)250 mph (1 point).

Direction: \(\tan^{-1}(\frac{70}{240})\approx\) \(16.3^{\circ}\) east of north (1 point).

Scoring: 1 point for the speed, 1 point for the angle.

Question 9: Two forces (2 points)

\(\vec{F_{1}}=\langle 20, 0\rangle\), \(\vec{F_{2}}=\langle 20\cos 120^{\circ},\ 20\sin 120^{\circ}\rangle=\langle -10,\ 10\sqrt{3}\rangle\) (1 point).

Resultant \(\langle 10,\ 10\sqrt{3}\rangle\), magnitude \(\sqrt{100+300}=\)20 pounds, direction \(\tan^{-1}(\sqrt{3})=\)\(60^{\circ}\) (1 point).

Scoring: 1 point for the components and sum, 1 point for magnitude and direction.

Scoring summary

How to use this precalculus vectors test answer key

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