In Exercises 11-25, find the component form of the vector v⃗ using the information given about i

step 1 The magnitude of B is 26, and we know it lies in the fourth quadrant. So in the fourth quadrant,

In Exercises 11-25, find the component form of the vector v⃗...

In Exercises $11-25$, find the component form of the vector $\vec{v}$ using the information given about its magnitude and direction. Give exact values. $\|\vec{v}\|=26 ;$ when drawn in standard position $\vec{v}$ lies in Quadrant IV and makes an angle measuring $\arctan \left(\frac{5}{12}\right)$ with the positive $x$ -axis

In Exercises $11-25$, find the component form of the vector $\vec{v}$ using the information given about its magnitude and direction. Give exact values.
$\|\vec{v}\|=26 ;$ when drawn in standard position $\vec{v}$ lies in Quadrant IV and makes an angle measuring $\arctan \left(\frac{5}{12}\right)$ with the positive $x$ -axis

Key Concepts

Vector Magnitude

This is the length of the vector and represents how long the vector is in the coordinate plane. It is calculated as the square root of the sum of the squares of its component values, and when given, it is used to scale the unit vector determined by the direction angle.

Direction Angle

This is the angle that the vector makes with the positive x-axis when drawn in standard position. It indicates the vector's orientation and is crucial for determining the signs and values of the components using trigonometric functions.

Component Form of a Vector

The component form represents a vector in terms of its horizontal (x) and vertical (y) components. These components are typically obtained by multiplying the vector’s magnitude by the cosine and sine of its direction angle, respectively.

Trigonometric Relationships

Cosine and sine functions relate the angles to the adjacent and opposite sides of a right triangle, respectively. For a vector, these functions are used to compute the x and y components, with the cosine giving the x-component and the sine giving the y-component.

Quadrant Considerations

The signs of the vector components depend on the quadrant in which the vector lies. In this context, being in Quadrant IV means that while the x-component (from cosine) is positive, the y-component (from sine) is negative, which affects the final computed values.

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