Problem 39 - Contemporary Precalculus

Question

Find the magnitude and direction angle of the vector \(\boldsymbol{v}\). $$\mathbf{v}=6 \mathbf{j}$$

Step-by-Step Solution

Answer

Answer: The magnitude of the vector \(\boldsymbol{v}\) is 6 and the direction angle is \(90^\circ\).

Step 1: Find the magnitude of the vector \(\boldsymbol{v}\)

The magnitude of a vector \(\boldsymbol{v}=a\boldsymbol{i}+b\boldsymbol{j}\) is given by \(\|\boldsymbol{v}\|=\sqrt{a^2+b^2}\). In this case, \(a=0\) and \(b=6\). Therefore, the magnitude of the vector \(\boldsymbol{v}\) is: $$\|\boldsymbol{v}\|=\sqrt{0^2+6^2}=\sqrt{36}=6.$$

Step 2: Find the direction angle of the vector \(\boldsymbol{v}\)

To find the direction angle \(\theta\), we can use the formula: $$\tan(\theta)=\frac{b}{a}$$ In this case, since \(a=0\) and \(b=6\): $$\tan(\theta)=\frac{6}{0}$$ Since it is impossible to divide by zero, this means that the vector is pointing straight upward along the j-direction, and the direction angle is: $$\theta=90^\circ$$ The magnitude of the vector \(\boldsymbol{v}\) is 6 and the direction angle is \(90^\circ\).

Key Concepts

Magnitude of a Vector

The magnitude of a vector quantifies how far and how forcefully we'd have to move in a straight line to get from the vector's starting point to its ending point. Imagine you are taking a step; the size of your step represents the magnitude of your movement. In the context of the given exercise, the vector \textbf{v} is described only by its component along the \textbf{j} axis (the vertical axis), which stands at 6 units.

To calculate the magnitude of vector \textbf{v}, which is signified as \( \|\boldsymbol{v}\| \), we apply the Pythagorean theorem to the vector's horizontal (a) and vertical (b) components. If a vector \( \boldsymbol{v} \) has components \(a\boldsymbol{i}+b\boldsymbol{j}\), the magnitude is found using \( \|\boldsymbol{v}\| = \sqrt{a^2 + b^2} \). In our problem, since the horizontal component (a) is 0 and the vertical component (b) is 6, the magnitude is simply the absolute value of the vertical component, resolving to 6.

Direction Angle of a Vector

While magnitude tells us how ‘big’ the vector is, the direction angle tells us where the vector is heading. To determine the direction of a vector in a plane, we often use the angle it makes with the positive direction of the horizontal axis. This is known as the direction angle and is typically denoted by \( \theta \).

The direction angle for a vector with components \(a\boldsymbol{i}+b\boldsymbol{j}\) is calculated using trigonometry, specifically the tangent function, where \( \tan(\theta) = \frac{b}{a} \). In the case of the vector \( \boldsymbol{v} \) with only a vertical component, \( \tan(\theta) \) proves to be undefined since it involves division by zero, which is mathematically impossible. This implies that the vector is aligned with the vertical axis. Hence, our vector's direction angle is precisely \(90^\text{o}\), suggesting it points straight up.

Trigonometry in Vectors

Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. It plays a significant role in the context of vectors, especially when we deal with the vector components and their resultant magnitude and direction. The trigonometric functions sin, cos, and tan are commonly used to relate the angles to their corresponding vector components.

For instance, if we know a vector's magnitude \( \|\boldsymbol{v}\| \) and its direction angle \( \theta \), we can find its horizontal and vertical components using \( a = \|\boldsymbol{v}\| \cdot \cos(\theta) \) and \( b = \|\boldsymbol{v}\| \cdot \sin(\theta) \). Conversely, if we have the components, we can use inverse trigonometric functions to find the direction angle, which was demonstrated in the previous section concerning the direction angle of the vector.

Vector Components

Vector components are the building blocks of a vector, often visualized as the legs of a right triangle, with the vector itself being the hypotenuse. They represent the vector's projection onto the axes of a coordinate system. A vector in two-dimensional space can be expressed as a combination of its horizontal (x-axis) and vertical (y-axis) components, indicated by \( \boldsymbol{i} \) and \( \boldsymbol{j} \) respectively.

Usually, a vector \( \boldsymbol{v} \) is written in the form \(a\boldsymbol{i}+b\boldsymbol{j}\), where 'a' and 'b' are the numerical representations of the components. In the exercise, we had \( \boldsymbol{v} = 0\boldsymbol{i} + 6\boldsymbol{j} \), signifying that the vector has a zero horizontal component and a vertical component of 6 units. These components provide insight into the directional influence of the vector within the plane.