Points P and Q have position vectors (2/b) and (8/13) respectively. Given that |PQ| = 10, calculate the values of b.

March 19, 2026Mathematics

ScanSolve AI Answer

March 19, 2026

Answer

$2 \pm 3\sqrt{5}$

Step 1: Position vectors (in m)
$\vec{OP} = \begin{pmatrix} 1 \\ 2 \end{pmatrix}, \quad \vec{OQ} = \begin{pmatrix} 3 \\ b \end{pmatrix}$

Step 2: Vector $\vec{PQ}$ (in m)
Formula: $\vec{PQ} = \vec{OQ} - \vec{OP}$
Substitute: $\vec{PQ} = \begin{pmatrix} 3 \\ b \end{pmatrix} - \begin{pmatrix} 1 \\ 2 \end{pmatrix}$
Simplify: $\vec{PQ} = \begin{pmatrix} 2 \\ b-2 \end{pmatrix}$

Step 3: Magnitude $|\vec{PQ}| = 7$ m
Formula: $|\vec{PQ}| = \sqrt{ (2)^2 + (b-2)^2 }$
Given: $\sqrt{ 4 + (b-2)^2 } = 7$

Step 4: Square both sides
Formula: $\left( \sqrt{ 4 + (b-2)^2 } \right)^2 = 7^2$
Substitute: $4 + (b-2)^2 = 49$
Simplify: $(b-2)^2 = 45$

Step 5: Solve for $b$ (in m)
Formula: $b-2 = \pm \sqrt{45}$
Substitute: $\sqrt{45} = \sqrt{9 \times 5} = 3\sqrt{5}$
Simplify: $b = 2 \pm 3\sqrt{5}$

$\mathbf{b = 2 \pm 3\sqrt{5} \, m}$
$\boxed{2 \pm 3\sqrt{5}}$

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Position vectors (in m) OP = 1 \\ 2 , OQ = 3 \\ b Step 2: Vector PQ (in m) Formula: PQ = OQ - OP Substitute: PQ = 3 \\ b - 1 \\ 2 Simplify: PQ = 2 \\ b-2 Step 3: Magnitude |PQ| = 7 m Formula: |PQ| = sqrt( (2)^2 + (b-2)^2 ) Given: sqrt( 4 + (b-2)^2 ) =…

Asked on March 19, 2026Mathematics

This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.

ScanSolve AI AnswerAnswered on March 19, 2026

Step 1: Position vectors (in m) OP = 1 \\ 2 , OQ = 3 \\ b Step 2: Vector PQ (in m) Formula: PQ = OQ - OP Substitute: PQ = 3 \\ b - 1 \\ 2 Simplify: PQ = 2 \\ b-2 Step 3: Magnitude |PQ| = 7 m Formula: |PQ| = sqrt( (2)^2 + (b-2)^2 ) Given: sqrt( 4 + (b-2)^2 ) = 7 Step 4: Square both sides Formula: ( sqrt( 4 + (b-2)^2 ) )^2 = 7^2 Substitute: 4 + (b-2)^2 = 49 Simplify: (b-2)^2 = 45 Step 5: Solve for b (in m) Formula: b-2 = ± sqrt(45) Substitute: sqrt(45) = sqrt(9 × 5) = 3sqrt(5) Simplify: b = 2 ± 3sqrt(5) b = 2 ± 3sqrt(5) \, m 2 ± 3sqrt(5)