Problem 4 - Calculus for Biology and Medicine

Question

In Problems , represent each given vector \(\mathbf{x}=\left[\begin{array}{l}x_{1} \\ x_{2}\end{array}\right]\) in the \(x_{1}-x_{2}\) plane, and determine its length and the angle that it forms with the positive \(x_{1}-\) axis (measured counterclockwise). $$ \mathbf{x}=\left[\begin{array}{r} -2 \\ 0 \end{array}\right] $$

Step-by-Step Solution

Answer

The vector lies on the negative \(x_1\)-axis, has length 2, and forms a 180-degree angle with the positive \(x_1\)-axis.

Step 1 - Plot the Vector

To represent the vector \(\mathbf{x}=\begin{bmatrix} -2 \ 0 \end{bmatrix}\) in the \(x_1-x_2\) plane, start by identifying its components. The vector starts at the origin \((0,0)\) and ends at the point \((-2,0)\). In a coordinate plane, this means draw a straight line from the origin to the point \((-2,0)\). The vector lies entirely on the negative \(x_1\)-axis (horizontal axis) since \(x_2 = 0\).

Step 2 - Calculate the Length of the Vector

The length or magnitude of the vector \(\mathbf{x}=\begin{bmatrix} -2 \ 0 \end{bmatrix}\) is given by the formula \(\|\mathbf{x}\| = \sqrt{x_1^2 + x_2^2}\). Substituting the values gives \(\|\mathbf{x}\| = \sqrt{(-2)^2 + 0^2} = \sqrt{4} = 2\).

Step 3 - Determine the Angle with the Positive \(x_1\)-Axis

The angle \(\theta\) that the vector \(\mathbf{x}\) forms with the positive \(x_1\)-axis can be found using \(\theta = \tan^{-1}\left(\frac{x_2}{x_1}\right)\). Substituting the components, \(\theta = \tan^{-1}\left(\frac{0}{-2}\right) = \tan^{-1}(0) = 0\) degrees. However, since the vector is on the negative \(x_1\)-axis, we must add 180 degrees to account for the direction change, resulting in an angle \(\theta = 180\) degrees counterclockwise from the positive \(x_1\)-axis.

Key Concepts

Magnitude of a Vector

The magnitude or length of a vector can be thought of as the distance from the starting point to the endpoint of the vector in the coordinate plane.
If you imagine the vector as a straight-line path, the magnitude is like measuring how long that path is. To find the magnitude of a vector \ \(\mathbf{x} = \begin{bmatrix} x_1 \ x_2 \end{bmatrix}\ \), you use the formula:

• \( \|\mathbf{x}\| = \sqrt{x_1^2 + x_2^2} \)

In the case of the vector \ \(\mathbf{x} = \begin{bmatrix} -2 \ 0 \end{bmatrix}\ \), the steps are:

• Square the components: \((-2)^2 = 4\) and \(0^2 = 0\)
• Add the squares: \(4 + 0 = 4\)
• Take the square root: \(\sqrt{4} = 2\)

Thus, the magnitude of this vector is 2. Think of it as the actual length of the vector from the point \ \((0,0)\ \) to \ \((-2,0)\ \) along the horizontal axis.

Angle with Axis

The angle a vector forms with an axis helps to understand its direction relative to standard positions in a plane.
For a vector \ \(\mathbf{x} = \begin{bmatrix} x_1 \ x_2 \end{bmatrix}\ \), the angle \(\theta\) with the positive \(x_1\)-axis is determined through the tangent function. You apply the formula:

• \( \theta = \tan^{-1}\left(\frac{x_2}{x_1}\right) \)

For our vector where \ \(x_1 = -2\ \) and \ \(x_2 = 0\ \), this becomes:

• \( \theta = \tan^{-1}\left(\frac{0}{-2}\right) = \tan^{-1}(0) = 0\text{ degrees} \)

However, since the vector lies on the negative \(x_1\)-axis, the correct interpretation is an angle of 180 degrees.
This accounts for the counterclockwise rotation needed to reach the vector from the positive side of the axis, illustrating the complete turn from the positive direction to the negative direction.

Coordinate Plane

The coordinate plane is a two-dimensional surface used to chart points and vectors using a system of axes. It consists of:

• The horizontal axis, often labeled as the \(x_1\)-axis
• The vertical axis, usually represented as the \(x_2\)-axis

Each point on the plane can be identified by a pair of coordinates \ \((x_1, x_2)\ \), indicating the point's position relative to the origin \ \((0,0)\ \).
In this exercise, the vector starts at the origin and ends at \ \((-2, 0)\ \).
That means, on the coordinate plane, a line is drawn from the origin moving towards the left, as negative values of \(x_1\) move in that direction.The coordinate plane is essential for visualizing and analyzing vector properties, such as direction and angle with respect to the axes, making it easier to grasp geometric and spatial relationships in mathematics.