Draw each of the following vectors, label an angle that specifies the vector's direction, and then find the vector's magnitude and direction. v = (14i - 11j) m/s

Verified step by step guidance

1. Step 1: Understand the vector components. The vector v is given as v = (14i - 11j) m/s, where 14i represents the x-component (horizontal direction) and -11j represents the y-component (vertical direction).
2. Step 2: Draw the vector on a coordinate system. Start at the origin (0, 0). Move 14 units to the right along the x-axis (positive direction) and then move 11 units downward along the y-axis (negative direction). Label the vector v and the angle θ it makes with the positive x-axis.
3. Step 3: Calculate the magnitude of the vector using the Pythagorean theorem. The magnitude |v| is given by the formula: $\sqrt{{14}^2+{11}^2}$. Substitute the values and simplify.
4. Step 4: Determine the direction of the vector. The angle θ can be calculated using the formula: $\theta =\tan -1^{-1}\left(\frac{-11}{14}\right)$. Note that the negative sign in the y-component indicates the vector points downward.
5. Step 5: Interpret the angle θ. Since the vector lies in the fourth quadrant (positive x-component and negative y-component), adjust the angle if necessary to ensure it is measured counterclockwise from the positive x-axis. Express the direction in degrees or radians as appropriate.

Key Concepts

Vector Representation

Vectors are quantities that have both magnitude and direction, represented in a coordinate system. In this case, the vector v = (14i - 11j) m/s can be visualized in a two-dimensional plane, where 'i' represents the x-component and 'j' represents the y-component. Understanding how to graphically represent vectors is essential for visualizing their direction and magnitude.

Magnitude of a Vector

The magnitude of a vector is a measure of its length, calculated using the Pythagorean theorem. For the vector v = (14i - 11j) m/s, the magnitude is found by taking the square root of the sum of the squares of its components: |v| = √(14² + (-11)²). This value provides a scalar quantity that represents the strength of the vector.

Direction of a Vector

The direction of a vector is often specified by the angle it makes with a reference axis, typically the positive x-axis. This angle can be calculated using the arctangent function: θ = arctan(y/x), where y and x are the vector's components. For the vector v = (14i - 11j) m/s, determining the angle helps in understanding how the vector is oriented in the coordinate system.