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

step 1 For vector W and Quadrant 1, we want to graph it writing component form and write it in I and J

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. $$\mathbf{w} \text { in } \mathrm{QI},|\mathbf{w}|=9.5, \theta_{r}=74.5^{\circ}$$

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.
$$\mathbf{w} \text { in } \mathrm{QI},|\mathbf{w}|=9.5, \theta_{r}=74.5^{\circ}$$

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Key Concepts

Vector Components

Vectors in a plane can be described using their horizontal and vertical components. The process involves decomposing the vector, which is initially defined by its magnitude and a specific angle relative to the horizontal axis, into contributions along the x-axis and y-axis. This is essential for converting between geometric and algebraic representations of vectors.

Trigonometric Decomposition

The trigonometric functions cosine and sine are used to determine the horizontal and vertical components of a vector, respectively. Specifically, the horizontal component is found by multiplying the magnitude by the cosine of the angle, while the vertical component is found by multiplying the magnitude by the sine of the angle. This relies on the fundamental definition of cosine and sine in a right triangle.

i, j Notation

Once the horizontal and vertical components of a vector are determined, the vector can be expressed in i, j notation. In this representation, the i unit vector corresponds to the horizontal (x) direction and the j unit vector corresponds to the vertical (y) direction. This notation provides a clear and concise way to write vectors in component form.

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