Problem 60 - Algebra and Trigonometry

Question

If \(P=(-3,1)\) and \(Q=(x, 4),\) find all numbers \(x\) so that the vector represented by \(\overline{P Q}\) has length 5 .

Step-by-Step Solution

Answer

x = 1 or x = -7

Step 1 - Understand the formula for vector length

The length of a vector \(\overline{P Q}\) can be found using the distance formula: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] where \(P(x_1, y_1) = (-3,1)\) and \(Q(x_2, y_2) = (x,4)\).

Step 2 - Substitute the given points into the distance formula

Substitute \(P = (-3,1)\) and \(Q = (x, 4)\) into the formula: \[ 5 = \sqrt{(x - (-3))^2 + (4 - 1)^2} \] which simplifies to: \[ 5 = \sqrt{(x + 3)^2 + 3^2} \]

Step 3 - Square both sides

To remove the square root, square both sides of the equation: \[ 5^2 = (x + 3)^2 + 3^2 \] This simplifies to: \[ 25 = (x + 3)^2 + 9 \]

Step 4 - Isolate the \( (x + 3)^2 \) term

Subtract 9 from both sides to isolate \( (x + 3)^2 \): \[ 25 - 9 = (x + 3)^2 \] which simplifies to: \[ 16 = (x + 3)^2 \]

Step 5 - Solve for \(x\)

To solve for \(x\), take the square root of both sides: \[ \pm\sqrt{16} = x + 3 \] which simplifies to: \[ \pm4 = x + 3 \] Therefore, \ x = 1 \ or \ x = -7 \.

Key Concepts

Distance Formula

The distance formula is a key concept when calculating the length of a vector between two points. It helps us find the straight-line distance (or length) between two points in a coordinate plane. The formula is derived from the Pythagorean theorem and is given as: