Question

Vector \(\overrightarrow{\mathbf{A}}\) is directed along the positive \(x\) -axis and has magnitude 1.73 units. Vector \(\overrightarrow{\mathbf{B}}\) is directed along the negative \(x\) -axis and has magnitude 1.00 unit. (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 for vectors \(\overrightarrow{\mathbf{A}}+\overrightarrow{\mathbf{B}}, \overrightarrow{\mathbf{A}}-\overrightarrow{\mathbf{B}} ,\) and \(\overrightarrow{\mathbf{B}}-\overrightarrow{\mathbf{A}}\) given that \(\overrightarrow{\mathbf{A}}\) is 1.73 units along the positive x-axis and \(\overrightarrow{\mathbf{B}}\) is 1.00 unit along the negative x-axis. Answer: The magnitudes and directions are as follows: (a) \(\overrightarrow{\mathbf{A}}+\overrightarrow{\mathbf{B}}\): Magnitude is 0.73 units, and the direction is along the positive x-axis. (b) \(\overrightarrow{\mathbf{A}}-\overrightarrow{\mathbf{B}}\): Magnitude is 2.73 units, and the direction is along the positive x-axis. (c) \(\overrightarrow{\mathbf{B}}-\overrightarrow{\mathbf{A}}\): Magnitude is 2.73 units, and the direction is along the negative x-axis.

Step-by-step solution

Part (a) Magnitude of \(\overrightarrow{\mathbf{A}}+\overrightarrow{\mathbf{B}}\)

Since \(\overrightarrow{\mathbf{A}}\) is directed along the positive x-axis, it is mandatory to write, \(\overrightarrow{\mathbf{A}} = 1.73\hat{\mathbf{i}}\). Similarly, Vector \(\overrightarrow{\mathbf{B}}\) is directed along the negative x-axis, so we get \(\overrightarrow{\mathbf{B}} = -1.00\hat{\mathbf{i}}\). Now to find \(\overrightarrow{\mathbf{A}}+\overrightarrow{\mathbf{B}}\), just sum the components of both vectors: \(\overrightarrow{\mathbf{A}}+\overrightarrow{\mathbf{B}} = (1.73 -1.00)\hat{\mathbf{i}} = 0.73\hat{\mathbf{i}}\). The magnitude of \(\overrightarrow{\mathbf{A}}+\overrightarrow{\mathbf{B}} =\) 0.73 units

Part (a) Direction of \(\overrightarrow{\mathbf{A}}+\overrightarrow{\mathbf{B}}\)

Since the resulting vector has a positive component in the x-axis, it is directed along the positive x-axis.

Part (b) Magnitude of \(\overrightarrow{\mathbf{A}}-\overrightarrow{\mathbf{B}}\)

To find \(\overrightarrow{\mathbf{A}}-\overrightarrow{\mathbf{B}}\), just subtract the components of both vectors: \(\overrightarrow{\mathbf{A}}-\overrightarrow{\mathbf{B}} = (1.73 - (-1.00))\hat{\mathbf{i}} = 2.73\hat{\mathbf{i}}\). The magnitude of \(\overrightarrow{\mathbf{A}}-\overrightarrow{\mathbf{B}} =\) 2.73 units

Part (b) Direction of \(\overrightarrow{\mathbf{A}}-\overrightarrow{\mathbf{B}}\)

Since the resulting vector has a positive component in the x-axis, it is directed along the positive x-axis.

Part (c) Magnitude of \(\overrightarrow{\mathbf{B}}-\overrightarrow{\mathbf{A}}\)

To find \(\overrightarrow{\mathbf{B}}-\overrightarrow{\mathbf{A}}\), just subtract the components of both vectors: \(\overrightarrow{\mathbf{B}}-\overrightarrow{\mathbf{A}} = (-1.00-1.73)\hat{\mathbf{i}} = -2.73\hat{\mathbf{i}}\). The magnitude of \(\overrightarrow{\mathbf{B}}-\overrightarrow{\mathbf{A}} =\) 2.73 units

Part (c) Direction of \(\overrightarrow{\mathbf{B}}-\overrightarrow{\mathbf{A}}\)

Since the resulting vector has a negative component in the x-axis, it is directed along the negative x-axis.