Problem 3 - Precalculus

Question

Find the magnitude of the vector \(\mathbf{A B} .\) $$A=(4,1) \text { and } B=(-3,0)$$

Step-by-Step Solution

Answer

The magnitude of vector \(\mathbf{A B}\) is \(5\sqrt{2}\).

Step 1: Identify the Components of Vector AB

First, determine the components of the vector \(\mathbf{A B}\) by subtracting the coordinates of point \(A\) from point \(B\). This gives us:\[ \mathbf{A B} = (B_x - A_x, B_y - A_y) = (-3 - 4, 0 - 1) = (-7, -1) \]

Step 2: Use the Magnitude Formula for Vectors

The magnitude of a vector \((x, y)\) is calculated using the formula:\[ \|\mathbf{A B}\| = \sqrt{x^2 + y^2} \]

Step 3: Substitute the Components and Calculate the Magnitude

Substitute the components of \(\mathbf{A B}\) into the magnitude formula:\[ \|\mathbf{A B}\| = \sqrt{(-7)^2 + (-1)^2} = \sqrt{49 + 1} = \sqrt{50} \]Simplify:\[ \|\mathbf{A B}\| = \sqrt{25 \times 2} = 5 \sqrt{2} \]

Key Concepts

Coordinate Subtraction

Coordinate subtraction is essential when working with vectors in a coordinate plane. When you want to find the vector from point \(A\) to point \(B\), you subtract the coordinates of \(A\) from \(B\). This mathematical operation gives you the vector components. For example, given two points \(A = (4, 1)\) and \(B = (-3, 0)\):

• Subtract the x-coordinates: \(-3 - 4 = -7\).
• Subtract the y-coordinates: \(0 - 1 = -1\).

The result is a new vector \((-7, -1)\), which represents the direction and the magnitude of the movement from \(A\) to \(B\).
By understanding and applying coordinate subtraction, you can accurately determine how far and in which direction one point is relative to another.

Magnitude Formula

The magnitude of a vector gives you the length or size of the vector, representing the distance from the initial to the terminal point. To calculate this, you utilize the magnitude formula for a two-dimensional vector \((x, y)\):
\[\|\mathbf{V}\| = \sqrt{x^2 + y^2}\]This formula is derived from the Pythagorean theorem, where the vector components \(x\) and \(y\) form the legs of a right triangle, and the magnitude is the hypotenuse.
By substituting the components calculated from coordinate subtraction (like \((-7, -1)\)), you'll calculate:
\[\|\mathbf{V}\| = \sqrt{(-7)^2 + (-1)^2} = \sqrt{49 + 1} = \sqrt{50}\]
Thus, the magnitude \( \sqrt{50} \) can be simplified to \(5\sqrt{2}\), providing a clearer understanding of the vector's length.

Vector Components

Vector components are crucial elements in describing a vector's direction and magnitude. These components arise from subtracting coordinates, shown in the exercise as \((-7, -1)\). Each component corresponds to movement along the specific axis:

• The horizontal component \((-7)\) indicates movement left on the x-axis since it's negative.
• The vertical component \((-1)\) shows a downward shift on the y-axis due to its negativity.

Understanding vector components helps in visualizing how vectors behave. By recognizing that each part acts independently, you can decompose complex movement into simple horizontal and vertical shifts. This makes analyzing and manipulating vectors more intuitive, especially in physics and engineering applications.