Problem 2 - Precalculus with Calculus Previews

Question

Sketch the given vector. Find the magnitude and the smallest positive direction angle of each vector. $$ \langle 4,-4\rangle $$

Step-by-Step Solution

Answer

Magnitude is \(4 \sqrt{2}\), direction angle is \(315^\circ\).

Step 1: Understand the Vector

The given vector \( \langle 4, -4 \rangle \) is written in component form, where 4 is its x-component and -4 is its y-component. This vector can be graphically represented in a coordinate system.

Step 2: Sketch the Vector

To sketch the vector \( \langle 4, -4 \rangle \), start at the origin \((0, 0)\). Move 4 units along the positive x-axis (right), then 4 units down along the negative y-axis. Draw an arrow from the origin to the point (4, -4) to represent the vector.

Step 3: Calculate the Magnitude

The magnitude of a vector \( \langle a, b \rangle \) is calculated using the formula: \[ \|\mathbf{v}\| = \sqrt{a^2 + b^2} \]Substitute the components:\[ \|\mathbf{v}\| = \sqrt{4^2 + (-4)^2} = \sqrt{16 + 16} = \sqrt{32} = 4\sqrt{2} \]

Step 4: Find the Direction Angle

The direction angle \( \theta \) of a vector is found using the arctangent of the y-component divided by the x-component: \[ \theta = \arctan\left(\frac{b}{a}\right) \]For \( \langle 4, -4 \rangle \), \[ \theta = \arctan\left(\frac{-4}{4}\right) = \arctan(-1) \]Since the vector is in the fourth quadrant, the angle is negative.\[ \theta = -45^\circ \]To find the smallest positive angle, add 360° to get \( 315° \).

Key Concepts

Vector Magnitude

The magnitude of a vector, also known as its length or size, is a measure of how long the vector is from where it starts to where it ends. For any vector given in component form, such as \( \langle a, b \rangle \), the magnitude \( \|\mathbf{v}\| \) is found using the Pythagorean theorem. The formula is:

• \( \|\mathbf{v}\| = \sqrt{a^2 + b^2} \)

To conceptualize this, imagine a right triangle where the vector forms the hypotenuse. The components \( a \) and \( b \) are the triangle's legs. The magnitude formula effectively calculates the hypotenuse length using the values of these legs. In our example, for the vector \( \langle 4, -4 \rangle \), substituting the values gives:

• \( \|\mathbf{v}\| = \sqrt{4^2 + (-4)^2} = \sqrt{16 + 16} = \sqrt{32} = 4\sqrt{2} \)

The magnitude tells you precisely how far the endpoint of the vector is from the origin in a straight line.

Direction Angle

The direction angle of a vector describes the orientation or angle the vector makes with the positive x-axis, effectively showing which direction the vector points. Calculating this angle involves understanding the vector's position relative to the coordinate plane.To find the direction angle \( \theta \) of a vector \( \langle a, b \rangle \):

• The formula used is \( \theta = \arctan\left(\frac{b}{a}\right) \).

For \( \langle 4, -4 \rangle \), the calculation of the angle is:

• \( \theta = \arctan\left(\frac{-4}{4}\right) = \arctan(-1) \).

This equation tells us the vector's angle relative to the x-axis. However, since \( \arctan(-1) = -45^\circ \), and considering the vector is in the fourth quadrant, a full circle adjustment to find the smallest positive angle is needed. Adding 360°, we find:

• \( 360^\circ + (-45^\circ) = 315^\circ \).

Thus, the smallest positive direction angle is \( 315^\circ \).

Component Form Vector

Vectors given in component form allow us to easily visualize and work with vectors in the coordinate plane. The component form is written as \( \langle a, b \rangle \), where \( a \) and \( b \) represent the vector's directional movement along the x and y axes, respectively.

• The x-component: Describes horizontal movement.
• The y-component: Describes vertical movement.

In our example, the vector \( \langle 4, -4 \rangle \) means starting at the origin, you would move 4 units to the right and 4 units down. This method is convenient for calculations, such as finding the magnitude or direction, since it breaks the vector into straightforward steps. Component form ensures any vector operation, such as addition, subtraction, and scaling, can be performed by simply manipulating these components individually. This is why understanding component form is fundamental in vector calculations.