Question

Displacement vector \(\overrightarrow{\mathbf{A}}\) is directed to the west and has magnitude \(2.56 \mathrm{km} .\) A second displacement vector is also directed to the west and has magnitude \(7.44 \mathrm{km}\) (a) What are the magnitude and direction of \(\overrightarrow{\mathbf{A}}+\overrightarrow{\mathbf{B}} ?\) (b) What are the magnitude and direction of \(\overrightarrow{\mathbf{A}}-\overrightarrow{\mathbf{B}} ?\) (c) What are the magnitude and direction of \(\overrightarrow{\mathbf{B}}-\overrightarrow{\mathbf{A}} ?\)

Short answer

Question: Calculate the magnitudes and directions of the following vector operations, given \(\overrightarrow{\mathbf{A}} = -2.56 \mathrm{km}\) west and \(\overrightarrow{\mathbf{B}} = -7.44 \mathrm{km}\) west: (a) \(\overrightarrow{\mathbf{A}}+\overrightarrow{\mathbf{B}}\) (b) \(\overrightarrow{\mathbf{A}}-\overrightarrow{\mathbf{B}}\) (c) \(\overrightarrow{\mathbf{B}}-\overrightarrow{\mathbf{A}}\) Answer: (a) Magnitude: \(10 \mathrm{km}\), Direction: West (b) Magnitude: \(4.88 \mathrm{km}\), Direction: East (c) Magnitude: \(4.88 \mathrm{km}\), Direction: West

Step-by-step solution

(a) Calculate \(\overrightarrow{\mathbf{A}}+\overrightarrow{\mathbf{B}}\)

Since both vectors are directed to the west, we can represent them as negative scalar values: \(\overrightarrow{\mathbf{A}} = -2.56 \mathrm{km}\) and \(\overrightarrow{\mathbf{B}} = -7.44 \mathrm{km}\) Adding them together, \(\overrightarrow{\mathbf{A}} + \overrightarrow{\mathbf{B}} = (-2.56) + (-7.44) = -10 \mathrm{km}\) The magnitude is \(10 \mathrm{km}\), and as the result is negative, the direction is to the west.

(b) Calculate \(\overrightarrow{\mathbf{A}}-\overrightarrow{\mathbf{B}}\)

Subtracting the vectors, \(\overrightarrow{\mathbf{A}} - \overrightarrow{\mathbf{B}} = (-2.56) - (-7.44) = 4.88 \mathrm{km}\) The magnitude is \(4.88 \mathrm{km}\), and as the result is positive, the direction is to the east.

(c) Calculate \(\overrightarrow{\mathbf{B}}-\overrightarrow{\mathbf{A}}\)

Subtracting the vectors in reverse order, \(\overrightarrow{\mathbf{B}} - \overrightarrow{\mathbf{A}} = (-7.44) - (-2.56) = -4.88 \mathrm{km}\) The magnitude is \(4.88 \mathrm{km}\), and as the result is negative, the direction is to the west.