Vectors in the Plane: Precalculus Math Test – PDF Download

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)

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.

See every chapter on the Precalculus math page.