Problem 34 - Precalculus : Building Concepts and Connections

Question

Find the magnitude and direction of each of the given vectors. Express the direction as an angle \(\theta\) in standard position, where \(0^{\circ} \leq \theta<360^{\circ},\) to tuo decimal places. $$\mathbf{v}=-3.2 \mathbf{i}+2 \mathbf{j}$$

Step-by-Step Solution

Answer

The magnitude of the vector is approximately 3.81 and the direction is approximately \(147.99^{\circ}\) in standard position.

Step 1: Find the magnitude

Magnitude of a vector \( \mathbf{v} = a\mathbf{i} + b\mathbf{j} \) can be found using the Pythagorean theorem as \( \sqrt{a^2 + b^2} \). For \(\mathbf{v} = -3.2\mathbf{i} + 2\mathbf{j}\), the magnitude \( |\mathbf{v}| \) would be \( \sqrt{(-3.2)^2 + 2^2} \) which equals to approximately 3.81.

Step 2: Find the direction

The direction or angle, denoted as \( \theta \), can be found using the tangent function as \( \theta = \arctan(\frac{b}{a}) \). Negatively, \(\theta = \arctan(\frac{2}{-3.2})\) which gives approximately -32.01 degrees due to arctan's range being \(-90^{\circ}<\theta<90^{\circ}\). Since the 'i' component is negative, and our angle is below the x-axis, we add 180 to it to get the correct angle, which in standard position is \(180 - 32.01 = 147.99\) degrees, rounded up to two decimal places.

Key Concepts

Understanding the Pythagorean Theorem

The Pythagorean theorem is a fundamental concept in mathematics, particularly when dealing with right-angled triangles. It states that in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.

Mathematically, if we have a triangle with sides of length 'a' and 'b', and the hypotenuse is 'c', then the Pythagorean theorem is expressed as:
\[ c^2 = a^2 + b^2 \]
In the context of vector magnitude, the components of the vector along the x-axis (\textbf{i}) and y-axis (\textbf{j}) act like the sides of a right-angled triangle, with the vector itself representing the hypotenuse. Hence, if \( \textbf{v} = a\textbf{i} + b\textbf{j} \), the magnitude of the vector is calculated using the same principle as the Pythagorean theorem:
\[ |\textbf{v}| = \sqrt{a^2 + b^2} \]
This approach allows us to determine the length or size of the vector, which is essential in understanding vector-based problems in physics, engineering, and mathematics.

Using the Tangent Function to Find Direction

The tangent function is a trigonometric ratio that compares two sides of a right triangle. Specifically, it is the ratio of the opposite side to the adjacent side. In a coordinate plane context, the tangent function helps find the angle that a vector makes with the positive direction of the x-axis.

Here’s how it is used:
- When we have a vector \( \textbf{v} = a\textbf{i} + b\textbf{j} \), the direction angle \( \theta \) of the vector can be found using the equation:\[ \theta = \arctan(\frac{b}{a}) \] - The inverse tangent function, \( \arctan \), is used to determine the angle whose tangent is \( \frac{b}{a} \).- However, since \( \arctan \) only gives values between \( -90^\circ \) and \( 90^\circ \), the actual direction angle must be adjusted based on the signs of 'a' and 'b' to place the angle in the correct quadrant of the Cartesian plane.

Adjusting for Quadrants

When the components 'a' or 'b' are negative, the vector will be situated in either the second or fourth quadrant, which means we have to compensate by adding or subtracting \( 180^\circ \) or \( 360^\circ \) accordingly to find the angle in standard position. This is vital for maintaining the correct orientation of the vector.

Defining the Standard Position Angle

The standard position angle of a vector is a clear way to express the orientation of the vector in the Cartesian coordinate system. It refers to the measure of the angle formed by the vector and the positive x-axis, starting from the axis and moving counterclockwise towards the vector's direction.

The angle is usually measured in degrees and by the following criteria:

• Angles are measured from the positive x-axis.
• Angles increase counterclockwise.
• If the vector is down and to the left (in the third or fourth quadrant), the angle is taken to be more than \( 180^\circ \) but less than \( 360^\circ \).

Whenever we calculate an angle using the tangent function, and the result is negative or does not fall into the desired range of \( 0^\circ \) to \( 360^\circ \) for the standard position angle, we adjust accordingly by adding \( 180^\circ \) to ensure the angle reflects the correct quadrant. This adjustment is crucial in vector analysis to maintain consistency and accuracy in representing vector directions.