Question

In Exercises 47–52, write the vector v in terms of i and j whose magnitude ||v|| and direction angle θ are given. ||v|| = 1/2, θ = 113°

Accepted Answer

Recall that a vector $$ \mathbf{v}  in $$the plane can be expressed in terms of the unit vectors $$ \mathbf{i} $$ and $$ \mathbf{j}  as$$ $$ \mathbf{v} = v_x \mathbf{i} + v_y \mathbf{j} $$, where $$ v_x $$ and $$ v_y $$ are the components of $$ \mathbf{v} $$ along the x- and y-axes respectively. Use the magnitude $$ ||\mathbf{v}|| $$ and the direction angle $$ 	heta  to $$find the components of $$ \mathbf{v} . $$The formulas for the components are:

$$ v_x = ||\mathbf{v}|| \cos(	heta) $$
$$ v_y = ||\mathbf{v}|| \sin(	heta) $$ Substitute the given values $$ ||\mathbf{v}|| = \frac{1}{2} $$ and $$ 	heta = 113^\circ $$ into the component formulas:

$$ v_x = \frac{1}{2} \cos(113^\circ) $$
$$ v_y = \frac{1}{2} \sin(113^\circ) $$ Write the vector $$ \mathbf{v}  in $$terms of $$ \mathbf{i} $$ and $$ \mathbf{j} $$ using the components found:

$$ \mathbf{v} = v_x \mathbf{i} + v_y \mathbf{j} $$ Remember that the cosine and sine of angles greater than 90° will be positive or negative depending on the quadrant, so consider the sign of each component based on the angle $$ 113^\circ $$ which lies in the second quadrant.

In Exercises 47–52, write the vector v in terms of i and j whose magnitude ||v|| and direction angle θ are given. ||v|| = 1/2, θ = 113°

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주요 개념

Vector Representation in the Plane

Magnitude and Direction Angle of a Vector

Using Trigonometry to Find Vector Components