Vectors in the Plane Test – Precalculus Math (Printable PDF)

Question 1 Vector Between Points (2 points)

Find the vector from P(−2, 6) to Q(5, −1). (1 point) Then find its magnitude to the nearest hundredth. (1 point)

Question 2 Components from a Direction (2 points)

A vector has magnitude 40 and direction angle 125°. Find its horizontal component (1 point) and its vertical component (1 point) to the nearest hundredth.

Question 3 Vector Sum from a Figure (2 points)

The figure shows u from the origin to (2, 5) and v drawn from the tip of u to the point (6, 2).

1. Read the components of u and v and find u + v. (1 point)
2. Find 2u − v. (1 point)

Question 4 Unit Vector (3 points)

Let v = ⟨−3, 7⟩. (a) Find the unit vector in the direction of v. (1 point) (b) Show that its magnitude is 1. (1 point) (c) Find the vector of length 20 in the same direction, to the nearest hundredth. (1 point)

Question 5 Dot Product and Angle (3 points)

Let u = ⟨5, −2⟩ and v = ⟨3, 5⟩. (a) Find u · v. (1 point) (b) Find |u| and |v|. (1 point) (c) Find the angle between them to the nearest tenth of a degree. (1 point)

Question 6 Kayak in a Current (3 points)

A kayaker paddles straight across a river (north) at 5.5 mph while the current flows east at 1.8 mph.

1. Write the resultant velocity vector. (1 point)
2. Find the actual speed to the nearest hundredth. (1 point)
3. Find the angle of the path east of north, to the nearest tenth of a degree. (1 point)

Question 7 Forces in Equilibrium (3 points)

Two forces act on a ring: F₁ = ⟨−35, 12⟩ pounds and F₂ = ⟨20, 26⟩ pounds, as shown.

1. Find the third force F₃ that puts the ring in equilibrium. (2 points)
2. Find the magnitude of F₃ to the nearest tenth. (1 point)

Question 8 Perpendicular Vectors (2 points)

Find k so that ⟨k, −5⟩ is perpendicular to ⟨6, 4⟩. Set up the condition first. (1 point) Then solve for k. (1 point)