Vectors in the Plane: Precalculus Math Test – PDF Download

Precalculus Vectors Test: questions 1–5

Question 1: Components and magnitude (2 points)

Points \(P(-2, 3)\) and \(Q(4, -5)\) are shown.

1. Write \(\vec{PQ}\) in component form. (1 point)
2. Find \(|\vec{PQ}|\). (1 point)

Question 2: Magnitude and direction (2 points)

Let \(\vec{v}=\langle -3,\ 3\sqrt{3}\rangle\).

1. Find \(|\vec{v}|\). (1 point)
2. Find the direction angle of \(\vec{v}\) from \(0^{\circ}\) to \(360^{\circ}\). (1 point)

Question 3: Combining vectors (2 points)

Let \(\vec{u}=\langle 2, -5\rangle\) and \(\vec{v}=\langle -4, 3\rangle\). Find each vector.

1. \(3\vec{u}+2\vec{v}\) (1 point)
2. \(\vec{u}-2\vec{v}\) (1 point)

Question 4: Parallelogram rule (3 points)

The figure shows vectors \(\vec{a}\) and \(\vec{b}\) drawn from the origin.

1. Read the components of \(\vec{a}\) and \(\vec{b}\), then find \(\vec{a}+\vec{b}\). (1 point)
2. Find \(\vec{a}-\vec{b}\). (1 point)
3. Find \(|\vec{a}+\vec{b}|\) exactly. (1 point)

Question 5: Unit vector (2 points)

Let \(\vec{v}=\langle -9, 12\rangle\).

1. Find \(|\vec{v}|\). (1 point)
2. Find the unit vector in the direction of \(\vec{v}\). (1 point)

Precalculus Vectors Test: questions 6–9

Question 6: Dot product and angle (3 points)

Let \(\vec{u}=\langle 2, 4\rangle\) and \(\vec{v}=\langle -1, 3\rangle\).

1. Find \(\vec{u}\cdot\vec{v}\). (1 point)
2. Find \(|\vec{u}|\) and \(|\vec{v}|\). (1 point)
3. Find the angle between \(\vec{u}\) and \(\vec{v}\). (1 point)

Question 7: Perpendicular vectors (2 points)

Find the value of \(k\) for which \(\langle k, 5\rangle\) is perpendicular to \(\langle 10, -4\rangle\). Show that the dot product is 0 for your value.

Question 8: Airplane and wind (2 points)

A plane has an airspeed of 240 mph heading due north. A wind blows due east at 70 mph (see figure). Find the plane’s ground speed and the direction of its path, measured east of north to the nearest tenth of a degree.

Question 9: Two forces (2 points)

Two forces of 20 pounds each act on an object. \(\vec{F_{1}}\) points along the positive x-axis and \(\vec{F_{2}}\) is at \(120^{\circ}\) from it (see figure). Find the magnitude and direction angle of the resultant force.

How to use this precalculus vectors test

Take this precalculus vectors test without notes, then grade it with the test answer key out of 20 points.

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