Question

name:______________________ mr. gurley - pre calculus january 16, 2026 vectors 6.5 word problems with linear combinations modeling tip: identify the basic vectors first. then write a linear combination of vectors use bracket form ⟨a, b⟩ or i-j form ai + bj for your final answer. 1. navigation: multiple legs of a walk a person walks with three moves. 2 east steps, then 3 north steps, then 1 west step. compute the resulting vector \boldsymbol{\vec{r}} and then find the magnitude |\vec{r}| - east step: \vec{e} = ⟨4, 0⟩ (blocks) - north step: \vec{n} = ⟨0, 3⟩ (blocks) 2. navigation: wind correction a drone intends to move with velocity \vec{v} = 8i + 15j \frac{meters}{sec} but a steady wind pushes with velocity \vec{w} = -2i + 1j \frac{meters}{sec}. compute the true speed \vec{r} and find the parallel unit vector \vec{r}. 3. soccer: two passes and a shot a player makes two passes and then takes a shot. model each action as a displacement vector on the field. compute the net displacement vector \vec{r} = \vec{p} + \vec{q} + \vec{s} (dont forget units!) first pass: \vec{p} = ⟨6, 8⟩ (m) second pass: \vec{q} = ⟨2, -3⟩ (m) shot on goal: \vec{s} = ⟨0, 4⟩ (m) page 1

Explanation:

Problem 1

Step1: Define west vector

West is opposite east, so $\vec{w} = \langle -4, 0
angle$ blocks

Step2: Sum all movement vectors

$\vec{r} = 2\vec{e} + 3\vec{n} + \vec{w} = 2\langle 4,0
angle + 3\langle 0,3
angle + \langle -4,0
angle$
$= \langle 8,0
angle + \langle 0,9
angle + \langle -4,0
angle = \langle 8+0-4, 0+9+0
angle = \langle 4,9
angle$

Step3: Calculate magnitude of $\vec{r}$

$|\vec{r}| = \sqrt{4^2 + 9^2}$

Step1: Find true velocity vector

$\vec{r} = \vec{v} + \vec{w} = (8i + 15j) + (-2i + 1j)$
$= (8-2)i + (15+1)j = 6i + 16j$ $\frac{\text{meters}}{\text{sec}}$

Step2: Calculate true speed (magnitude)

$|\vec{r}| = \sqrt{6^2 + 16^2} = \sqrt{36 + 256} = \sqrt{292} = 2\sqrt{73}$

Step3: Find parallel unit vector

$\hat{r} = \frac{\vec{r}}{|\vec{r}|} = \frac{6i + 16j}{2\sqrt{73}} = \frac{3}{\sqrt{73}}i + \frac{8}{\sqrt{73}}j$ (or rationalized: $\frac{3\sqrt{73}}{73}i + \frac{8\sqrt{73}}{73}j$)

Step1: Sum all displacement vectors

$\vec{r} = \vec{p} + \vec{q} + \vec{s} = \langle 6,8
angle + \langle 2,-3
angle + \langle 0,4
angle$
$= \langle 6+2+0, 8+(-3)+4
angle = \langle 8,9
angle$

Answer:

Resulting vector: $\langle 4,9
angle$ blocks
Magnitude: $\sqrt{97}$ blocks

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Problem 2