For each vector here, θr represents the acute angle formed by the vector and the x -axis. (a) Gr

step 1 In this question, given beta r is equal to 32 .63, x -axis. And our vector is V is equal to mari

For each vector here, θr represents the acute angle formed...

For each vector here, $\theta_{r}$ represents the acute angle formed by the vector and the $x$ -axis. (a) Graph each vector, (b) find the horizontal and vertical components and write the vector in component form, and (c) write the vector in $\mathbf{i}, \mathbf{j}$ form. Round to the nearest tenth. $$\mathbf{v} \text { in } \mathrm{QIV},|\mathbf{v}|=20, \theta_{r}=32.6^{\circ}$$

For each vector here, $\theta_{r}$ represents the acute angle formed by the vector and the $x$ -axis. (a) Graph each vector, (b) find the horizontal and vertical components and write the vector in component form, and (c) write the vector in $\mathbf{i}, \mathbf{j}$ form. Round to the nearest tenth.
$$\mathbf{v} \text { in } \mathrm{QIV},|\mathbf{v}|=20, \theta_{r}=32.6^{\circ}$$

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Key Concepts

Vector Magnitude and Direction

This concept involves understanding that every vector has a magnitude (length) and a direction, typically given as an angle measured from the positive x-axis. The magnitude provides the scale of the vector, while its direction indicates where it points in the plane.

Conversion from Polar to Rectangular Coordinates

This conversion takes a vector given in polar form, with a magnitude and an angle, and determines its horizontal (x) and vertical (y) components using trigonometric relationships: x = r cos(?) and y = r sin(?). This process is fundamental in moving between polar and Cartesian representations.

Quadrantal Sign Considerations

When converting vectors from polar to rectangular form, it is important to consider the quadrant in which the vector lies. The signs of the computed x and y components depend on the quadrant; for instance, in the fourth quadrant, the horizontal component is positive and the vertical component is negative.

Component Form and Unit Vector Representation

The component form of a vector expresses it in terms of its horizontal and vertical parts, often written as an ordered pair or as multiples of the standard unit vectors i and j. This form simplifies operations like addition and subtraction of vectors by breaking them down into their components.

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