Problem 39 - Precalculus

Question

Find a unit vector in the direction of the given vector. $$\mathbf{v}=(24,-7)$$

Step-by-Step Solution

Answer

The unit vector in the direction of \( \mathbf{v} = (24, -7) \) is \( \left( \frac{24}{25}, -\frac{7}{25} \right) \)."}

Step 1: Analyze the problem

Identify the type of problem and select the appropriate mathematical approach.

Step 2: Solve step by step

The unit vector in the direction of \( \mathbf{v} = (24, -7) \) is \( \left( \frac{24}{25}, -\frac{7}{25} \right) \)."}.

Step 3: Verify

Check the solution by substitution or alternative method.

Key Concepts

Vector Magnitude

When we talk about the magnitude of a vector, we are essentially discussing how long the vector is. Magnitude represents the size or length of a vector.
For a two-dimensional vector \( \mathbf{v} = (x, y) \), the magnitude is calculated using the Pythagorean theorem. This is because a vector can be thought of as the hypotenuse of a right triangle, with its components \( x \) and \( y \) forming the other two sides.
The formula for the magnitude \( \| \mathbf{v} \| \) is:

• \( \| \mathbf{v} \| = \sqrt{x^2 + y^2} \)

In the context of our exercise, the vector \( \mathbf{v} = (24, -7) \) means we have:

• First component (x): 24
• Second component (y): -7

Plug these values into the formula, square the components, add them, and finally take the square root. This will give you the magnitude of the vector.

Vector Direction

Vectors not only have a magnitude, but they also have a direction. The direction of a vector is the angle it forms with a reference line, usually the positive x-axis in two-dimensional space.
This direction is especially significant in problems where the orientation of a force or movement is needed. In our case, identifying the direction helps in forming a unit vector in the same path as the original vector \( \mathbf{v} = (24, -7) \).
Remember, a unit vector is a vector that has a magnitude of exactly 1 but points in the same direction as the original vector. To find the unit vector's specific direction, we might calculate the angle using trigonometric functions, though in many cases it is enough to understand its pathway for computation without specifically measuring the angle.

Magnitude Calculation

Calculating the magnitude of a vector involves several steps. First, ensure you have all the vector's components; these are usually given in the format \( (x, y) \) for two dimensions.
For the vector \( \mathbf{v} = (24, -7) \), calculate the square of each component:

• \( x^2 = 24^2 = 576 \)
• \( y^2 = (-7)^2 = 49 \)

Then, add these squares together:

• \( 576 + 49 = 625 \)

Finally, take the square root of this sum to get the magnitude:

• \( \| \mathbf{v} \| = \sqrt{625} = 25 \)

Thus, the magnitude of vector \( \mathbf{v} \) is 25. This value is essential for further computations, like finding the unit vector in the same direction.