Question

The resultant vector C of vectors A and B is perpendicular to vector A. Also magnitudes of vectors A and C are equal. Find the angle between the vectors A and B.

Answer 1

Answer on Question #55643 – Math – Vector Calculus

The resultant vector $C$ of vectors $A$ and $B$ is perpendicular to vector $A$. Also magnitudes of vectors $A$ and $C$ are equal. Find the angle between the vectors $A$ and $B$.

Solution

Resulting vector of $\mathbf{A}$ and $\mathbf{B}$ is vector $\mathbf{C}$, which is perpendicular to $\mathbf{A}$, and $|\mathbf{A}| = |\mathbf{C}|$.

We represent the vector $\mathbf{B} = \mathbf{B}_1 + \mathbf{B}_2$, where $\mathbf{B}_1 = -\mathbf{A}$, $\mathbf{B}_2 = \mathbf{C}$, then

$$\mathbf{A} + \mathbf{B} = \mathbf{A} - \mathbf{A} + \mathbf{C} = \mathbf{C}.$$$$|\mathbf{B}| = \sqrt{\mathbf{B}_1^2 + \mathbf{B}_2^2} = \sqrt{\mathbf{A}^2 + \mathbf{C}^2} = \sqrt{2} |\mathbf{A}|,$$

Triangle, formed by vectors $\mathbf{B}_1, \mathbf{B}_2$ and $\mathbf{B}$, is right and isosceles, hence the measure of the angle between $\mathbf{B}_1$ and $\mathbf{B}$ is 45 degrees.

$\mathbf{A}$ is perpendicular to $\mathbf{C}$, so the measure of the angle between $\mathbf{A}$ and $\mathbf{C}$ is 90 degrees. Thus, the angle between vectors $\mathbf{B}$ and $\mathbf{C}$ is the difference of the right angle and the angle between $\mathbf{B}_1$ and $\mathbf{B}$, that is, measure is $90 - 45 = 45$ degrees. Now the angle between vectors $\mathbf{A}$ and $\mathbf{B}$ is the sum of the angle between $\mathbf{A}$ and $\mathbf{C}$ and the angle between $\mathbf{B}$ and $\mathbf{C}$, hence measure is $90 + 45 = 135$ degrees.

Answer: 135 degrees.

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