The displacement vectors 𝐀 and 𝐁 shown in Figure 𝐏 3.9 both...

The displacement vectors $\overrightarrow{\mathbf{A}}$ and $\overrightarrow{\mathbf{B}}$ shown in Figure $\mathbf{P} 3.9$ both have magnitudes of $3.00 \mathrm{~m}$. The direction of vec- tor $\overrightarrow{\mathbf{A}}$ is $\theta=30.0^{\circ}$. Find gra- (b) $\overrightarrow{\mathbf{A}}-\overrightarrow{\mathbf{B}}$, (c) $\overrightarrow{\mathbf{B}}-\overrightarrow{\mathbf{A}},$ and (d) $\overrightarrow{\mathbf{A}}-2 \overrightarrow{\mathbf{B}}$. (Report all angles counterclockwise from the positive $x$ axis.)

The displacement vectors $\overrightarrow{\mathbf{A}}$ and $\overrightarrow{\mathbf{B}}$ shown in Figure $\mathbf{P} 3.9$
both have magnitudes of $3.00 \mathrm{~m}$. The direction of vec-
tor $\overrightarrow{\mathbf{A}}$ is $\theta=30.0^{\circ}$. Find gra-
(b) $\overrightarrow{\mathbf{A}}-\overrightarrow{\mathbf{B}}$,
(c) $\overrightarrow{\mathbf{B}}-\overrightarrow{\mathbf{A}},$ and
(d) $\overrightarrow{\mathbf{A}}-2 \overrightarrow{\mathbf{B}}$.
(Report all angles counterclockwise from the positive $x$ axis.)

Key Concepts

Displacement Vector

A displacement vector represents a change in position that has both magnitude (or length) and direction. It describes the straight-line distance and the direction from an initial point to a final point, making it a fundamental concept in physics and engineering to represent motion and spatial relations.

Component Form of Vectors

Representing vectors in component form involves breaking them down into horizontal (x-axis) and vertical (y-axis) components. This is typically done using trigonometric functions, such as cosine for the x-component and sine for the y-component, given the vector's angle measured from the positive x-axis. This method simplifies calculations for vector addition, subtraction, and scalar multiplication.

Vector Addition and Subtraction

Vector addition and subtraction are operations that combine vectors by adding or subtracting their corresponding components. In these operations, the resultant vector is found by performing arithmetic on the x and y components separately. Subtraction, in particular, can be seen as adding a negative vector, which reverses the direction of the vector being subtracted.

Scalar Multiplication

Scalar multiplication involves multiplying a vector by a scalar (a real number). This operation scales the magnitude of the vector by the absolute value of the scalar while keeping its direction unchanged if the scalar is positive, or reversing its direction if the scalar is negative. Scalar multiplication is essential when scaling vectors in equations and transformations.

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