Question

In Exercises 43–44, find the angle, in degrees, between v and w.v = 2 cos(4π/3) i + 2 sin(4π/3) j, w = 3 cos(3π/2) i + 3 sin(3π/2) j

Accepted Answer

Identify the vectors $$ \mathbf{v} = 2 \cos(4\pi) \mathbf{i} + 2 \sin(4\pi) \mathbf{j} $$ and $$ \mathbf{w} = 3 \cos(3\pi) \mathbf{i} + 3 \sin(3\pi) \mathbf{j} . $$These are given in component form using trigonometric functions. Recall that the angle $$ 	heta $$ between two vectors $$ \mathbf{v} $$ and $$ \mathbf{w} $$ can be found using the dot product formula:

$$ \mathbf{v} \cdot \mathbf{w} = \|\mathbf{v}\| \|\mathbf{w}\| \cos(	heta) $$

which rearranges to

$$ 	heta = \cos^{-1} \left( \frac{\mathbf{v} \cdot \mathbf{w}}{\|\mathbf{v}\| \|\mathbf{w}\|} ight) . $$Calculate the components of $$ \mathbf{v} $$ and $$ \mathbf{w}  by $$evaluating the cosine and sine values at the given angles:

- $$ \cos(4\pi) $$ and $$ \sin(4\pi) $$
- $$ \cos(3\pi) $$ and $$ \sin(3\pi) . $$Compute the dot product $$ \mathbf{v} \cdot \mathbf{w} $$ using the formula:

$$ \mathbf{v} \cdot \mathbf{w} = v_x w_x + v_y w_y $$

where $$ v_x, v_y $$ and $$ w_x, w_y $$ are the components of $$ \mathbf{v} $$ and $$ \mathbf{w} $$ respectively. Find the magnitudes $$ \|\mathbf{v}\| $$ and $$ \|\mathbf{w}\| $$ using:

$$ \|\mathbf{v}\| = \sqrt{v_x^2 + v_y^2} \quad 	ext{and} \quad \|\mathbf{w}\| = \sqrt{w_x^2 + w_y^2} $$

Finally, substitute these values into the formula for $$ 	heta $$ and solve for the angle in degrees.

In Exercises 43–44, find the angle, in degrees, between v and w.
v = 2 cos(4π/3) i + 2 sin(4π/3) j, w = 3 cos(3π/2) i + 3 sin(3π/2) j

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주요 개념

Vector Representation in the Plane

Dot Product and Angle Between Vectors

Trigonometric Functions and Angle Measures