Vectors are essential tools for describing quantities that have both magnitude and direction, such as velocity, force, and displacement.
A delivery truck travels from a warehouse at point [latex]W(-3, 2)[/latex] to a store at point [latex]S(4, 5)[/latex], where coordinates represent city blocks. Find the position vector representing this displacement and graph both the original vector and position vector.
To find the position vector, we subtract the initial point coordinates from the terminal point coordinates: [latex]\begin{aligned} \mathbf{v} &= \langle 4 - (-3), 5 - 2 \rangle \ &= \langle 7, 3 \rangle \end{aligned}[/latex] The position vector [latex]\mathbf{v} = \langle 7, 3 \rangle[/latex] starts at the origin [latex](0, 0)[/latex] and ends at the point [latex](7, 3)[/latex]. This tells us the truck travels 7 blocks east and 3 blocks north.
A search and rescue helicopter needs to fly from its base at point [latex]P(-8, 1)[/latex] to a hiker’s location at point [latex]Q(-2, -5)[/latex], where coordinates are in kilometers. Find the magnitude and direction of the displacement vector.
First, find the position vector: [latex]\begin{aligned} \mathbf{u} &= \langle -2 - (-8), -5 - 1 \rangle \ &= \langle 6, -6 \rangle \end{aligned}[/latex] Next, calculate the magnitude using the Pythagorean Theorem: [latex]\begin{aligned} |\mathbf{u}| &= \sqrt{6^2 + (-6)^2} \ &= \sqrt{36 + 36} \ &= \sqrt{72} \ &= 6\sqrt{2} \ &\approx 8.49 \text{ km} \end{aligned}[/latex] Now find the direction angle: [latex]\begin{aligned} \tan \theta &= \frac{-6}{6} = -1 \ \theta &= \tan^{-1}(-1) = -45° \end{aligned}[/latex] Since the vector [latex]\langle 6, -6 \rangle[/latex] has a positive x-component and negative y-component, it terminates in the fourth quadrant. To express this as a positive angle measured counterclockwise from the positive x-axis: [latex]\theta = -45° + 360° = 315°[/latex] The helicopter needs to fly approximately [latex]8.49[/latex] km at a bearing of [latex]315°[/latex] (or equivalently, [latex]45°[/latex] south of due east).