Problem 6 - Precalculus

Question

Find the magnitude of the vector \(\mathbf{A B} .\) $$A=(-2,1) \text { and } B=(4,9)$$

Step-by-Step Solution

Answer

The magnitude of the vector \( \mathbf{AB} \) is 10.

Step 1: Identifying the formula

The magnitude of a vector \( \mathbf{AB} \) can be found using the distance formula: \[ ||\mathbf{AB}|| = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] where \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of points \(A\) and \(B\).

Step 2: Substituting the values

Plug the coordinates of points \(A(-2,1)\) and \(B(4,9)\) into the formula. \[ ||\mathbf{AB}|| = \sqrt{(4 - (-2))^2 + (9 - 1)^2} \] Which simplifies to: \[ ||\mathbf{AB}|| = \sqrt{(4 + 2)^2 + (9 - 1)^2} \]

Step 3: Calculating differences

Perform the calculations inside the square root. \[ 4 + 2 = 6 \] and \[ 9 - 1 = 8 \] Therefore, we have: \[ ||\mathbf{AB}|| = \sqrt{6^2 + 8^2} \]

Step 4: Calculating squares

Now, calculate the squares of the differences. \[ 6^2 = 36 \] and \[ 8^2 = 64 \] So, substitute these back into the expression: \[ ||\mathbf{AB}|| = \sqrt{36 + 64} \]

Step 5: Summing the squares

Add the squared terms together. \[ 36 + 64 = 100 \] Thus, \[ ||\mathbf{AB}|| = \sqrt{100} \]

Step 6: Finding the square root

Find the square root of the sum to get the magnitude. \[ \sqrt{100} = 10 \] Therefore, the magnitude of vector \( \mathbf{AB} \) is 10.

Key Concepts

Distance Formula

The distance formula is a mathematical expression to calculate the distance between two points in a plane. This is particularly useful in fields like geometry and physics. It is derived from the Pythagorean theorem and provides a way to measure the straight line between two points in a coordinate system. For instance, if you have two points with coordinates

• A (x_1, y_1)
• B (x_2, y_2)

then the distance between them, which is also the magnitude of the vector connecting them, can be calculated using:\[ \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]This formula works by considering each difference

• \( x_2 - x_1 \)
• \( y_2 - y_1 \)

as the sides of a right triangle and uses the hypotenuse to find the distance.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is the study of geometry using a coordinate system. This branch of mathematics uses numerical coordinates to represent geometric figures and analyze their properties. Points are often defined by pairs or triples of numbers. Some useful concepts in coordinate geometry include:

• Distance and midpoint formulas which help in finding the distance or the center point between two coordinates.
• Slope of a line that defines its steepness and direction.
• Equations of lines and curves that help in understanding their graphic representation.

In our exercise, both points A and B have coordinates that allow us to precisely calculate the vector's magnitude between them, essential in both math problems and real-world scenarios like navigation or map plotting.

Vector Calculations

Vector calculations are essential in understanding physical concepts like force, velocity, and movement. A vector carries both a direction and magnitude. When dealing with vectors in a plane, you often need to compute their magnitude, direction, or resultant vectors. In our exercise, the vector from point A to B is described in terms of its start and end coordinates. The magnitude of a vector, or its length, is what we calculate using the distance formula. To dive deeper into vector calculations, remember:

• Vectors can be added or subtracted following specific rules, often visualized as 'head-to-tail' methods.
• The direction of a vector can be described using angles or unit vectors.
• Vectors can be multiplied by scalars to change their magnitude without altering the direction.

By mastering these vector operations, students can solve complex problems in physics and engineering, providing clearer insights into hundreds of real-world applications.