Adding and Subtracting Vectors in the Plane

Identify whether a wind blowing at 15 knots from the northwest is a scalar or a vector.

It is a vector because it has both a magnitude (15 knots) and a specified direction (from the northwest).

Given vector a = (3, 5) and vector b = (2, 1), find the component form of vector a + vector b.

(5, 6)

Given vector u = (-4, 7) and vector v = (6, -3), calculate the sum u + v.

(2, 4)

Find the magnitude of the vector v = (6, 8).

10 units. Magnitude = sqrt(6^2 + 8^2) = sqrt(36 + 64) = sqrt(100) = 10.

Vector p has components (5, -2). Write the components of the opposite vector, -p.

(-5, 2)

Subtract vector w = (3, 9) from vector z = (7, 4). Calculate z - w.

(4, -5). z - w = (7 - 3, 4 - 9) = (4, -5).

A plane flies due east for 120 miles, then turns due north for 50 miles. What is the total straight-line displacement from the start?

130 miles at an angle of roughly 22.6 degrees north of east. Magnitude = sqrt(120^2 + 50^2) = sqrt(14400 + 2500) = sqrt(16900) = 130 miles.

Vectors m = (1, k) and n = (4, -2) add up to a resultant vector with a magnitude of 5 along the positive x-axis, so m + n = (5, 0). Find the value of k.

k = 2. The vertical components must sum to 0: k + (-2) = 0, so k = 2.