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|| = 12, θ = 225°

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 the vector along the x-axis and y-axis respectively. Use the magnitude $$ ||\mathbf{v}|| = 12 $$ and the direction angle $$ 	heta = 225^\circ  to $$find the components. The formulas for the components are:

$$ v_x = ||\mathbf{v}|| \cos(	heta) $$
$$ v_y = ||\mathbf{v}|| \sin(	heta) $$ Substitute the given values into the component formulas:

$$ v_x = 12 \cos(225^\circ) $$
$$ v_y = 12 \sin(225^\circ) $$ Evaluate the cosine and sine of $$ 225^\circ . $$Remember that $$ 225^\circ  is in $$the third quadrant where both sine and cosine are negative. Use the reference angle $$ 225^\circ - 180^\circ = 45^\circ  to $$find the exact values. Write the vector $$ \mathbf{v}  in $$terms of $$ \mathbf{i} $$ and $$ \mathbf{j} $$ using the components found:

$$ \mathbf{v} = v_x \mathbf{i} + v_y \mathbf{j} $$

In Exercises 47–52, write the vector v in terms of i and j whose magnitude ||v|| and direction angle θ are given. ||v|| = 12, θ = 225°

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Key Concepts

Vector Representation in the Plane

Magnitude and Direction Angle of a Vector

Converting Polar Form to Cartesian Components