Problem 13 - Precalculus

Question

Find the magnitude and direction angle of the given vector. $$\mathbf{u}=\langle-8,0\rangle$$

Step-by-Step Solution

Answer

Magnitude is 8, direction angle is 180°.

Step 1: Understand Magnitude

The magnitude of a vector \(\mathbf{u} = \langle a, b \rangle\) is computed using the formula \(\lVert\mathbf{u}\rVert = \sqrt{a^2 + b^2}\). This gives us the length of the vector from the origin to the point \(\langle a, b \rangle\).

Step 2: Calculate Magnitude

Substitute the components of the vector \(\mathbf{u} = \langle -8, 0 \rangle\) into the magnitude formula: \(\lVert\mathbf{u}\rVert = \sqrt{(-8)^2 + 0^2} = \sqrt{64} = 8\). Therefore, the magnitude is \(8\).

Step 3: Understand Direction Angle

The direction angle \(\theta\) of a vector \(\mathbf{u} = \langle a, b \rangle\) can be found using \(\theta = \tan^{-1} \left(\frac{b}{a}\right)\). This calculation helps determine the angle made by the vector with the positive x-axis.

Step 4: Calculate Direction Angle

For the vector \(\langle -8, 0 \rangle\), since the y-component \((b)\) is 0 and the x-component \((a)\) is negative, the vector points along the negative x-axis. Therefore, the angle \(\theta\) is \(180^\circ\), which means it points directly left.

Key Concepts

Magnitude of a Vector

The magnitude of a vector is an essential concept in understanding vector quantities. Essentially, the magnitude tells us how long the vector is, similar to the length of a line segment.
For any vector \(\mathbf{u} = \langle a, b \rangle\), its magnitude \(\lVert \mathbf{u} \rVert \) is calculated using the formula:\[\lVert \mathbf{u} \rVert = \sqrt{a^2 + b^2}\]
To put it simply:

• "\(a\)" and "\(b\)" are the components of the vector, representing its horizontal and vertical projections respectively.
• The formula is reminiscent of the Pythagorean theorem, where the magnitude is the hypotenuse of a right triangle formed by \(a\) and \(b\).

For example, for the vector \(\mathbf{u} = \langle -8, 0 \rangle\), the magnitude is \(\sqrt{(-8)^2 + 0^2} = \sqrt{64} = 8\).
In geometrical terms, this can be visualized as the straight-line distance from the origin to the point \(\langle -8, 0 \rangle \) on a graph.

Direction Angle of a Vector

The direction angle of a vector provides the orientation of the vector in a plane, specifically the angle it forms with the positive x-axis.
For any vector \(\mathbf{u} = \langle a, b \rangle\), the direction angle \(\theta\) can be determined using the tangent function, calculated as \(\theta = \tan^{-1} \left(\frac{b}{a}\right)\).
Here are key points to understand:

• \(b\) is the vertical component, often associated with the change in y.
• \(a\) is the horizontal component, associated with the change in x.
• The inverse tangent function \(\tan^{-1}\) gives the angle in radians or degrees.

For the given vector \(\langle -8, 0 \rangle\), since \(b = 0\) and \(a = -8\) (a negative value), the vector points leftward along the negative x-axis.
Thus, the direction angle is \(180^\circ\). This result aligns with the vector's orientation in the coordinate system, pointing directly to the left.

Vector Components

Understanding vector components is crucial for visualizing and solving vector problems.
A vector is often represented in a coordinate system by its components, which indicate how far and in which direction the vector extends in each dimension.
For a vector \(\mathbf{u} = \langle a, b \rangle\):

• The first component, \(a\), is the horizontal component, indicating movement along the x-axis.
• The second component, \(b\), is the vertical component, indicating movement along the y-axis.

These components allow us to break the vector into understandable parts. For instance, the vector \(\langle -8, 0 \rangle\) has a horizontal part that extends 8 units to the left and no vertical movement. In physical terms, this can be seen as an arrow on the graph spanning leftwards on the x-axis, starting at the origin (0,0) and ending at (-8,0).
This simple breakdown helps in visualizing vector operations and understanding their resultant displacement.