Number and Quantity · Lesson 6

Vectors

The vector from an initial point P(x1,y1)P(x_1, y_1) to a terminal point Q(x2,y2)Q(x_2, y_2) is ⟨x2−x1,y2−y1⟩\langle x_2 - x_1, y_2 - y_1 \rangle: terminal minus initial. In i\mathbf{i}, j\mathbf{j} notation, where i=⟨1,0⟩\mathbf{i} = \langle 1, 0 \rangle and j=⟨0,1⟩\mathbf{j} = \langle 0, 1 \rangle, the vector ⟨a,b⟩\langle a, b \rangle is written ai+bja\mathbf{i} + b\mathbf{j}.

For a combination such as u−2v\mathbf{u} - 2\mathbf{v}, scale first, then subtract component by component. The magnitude of ⟨a,b⟩\langle a, b \rangle is a2+b2\sqrt{a^2 + b^2}, and dividing each component by the magnitude gives a unit vector: length 1, same direction.

A trip with several legs is vector addition. Write each leg as a vector, with east and north positive and west and south negative, add them, and take the magnitude of the total to get the distance from the start. Two nonzero vectors are parallel when one is a scalar multiple of the other, so their components are in the same ratio.

Worked example Easy

Vector w\mathbf{w} has initial point P(2,−1)P(2, -1) and terminal point Q(7,3)Q(7, 3). Which of the following is w\mathbf{w} in component form?

  1. ⟨−5,−4⟩\langle -5, -4 \rangle
  2. ⟨5,2⟩\langle 5, 2 \rangle
  3. ⟨9,2⟩\langle 9, 2 \rangle
  4. ⟨5,4⟩\langle 5, 4 \rangleAnswer

How to solve it

  1. Terminal minus initial: ⟨7−2,3−(−1)⟩\langle 7 - 2, 3 - (-1) \rangle.
  2. Compute each component: 7−2=57 - 2 = 5 and 3+1=43 + 1 = 4, so w=⟨5,4⟩\mathbf{w} = \langle 5, 4 \rangle.

Why each choice is right or wrong

  • A. Incorrect. ⟨−5,−4⟩\langle -5, -4 \rangle is P−QP - Q, initial minus terminal. It has the right length but points the opposite way.
  • B. Incorrect. ⟨5,2⟩\langle 5, 2 \rangle drops the sign on −1-1 and computes 3−13 - 1. Subtracting −1-1 adds 1: 3+1=43 + 1 = 4.
  • C. Incorrect. ⟨9,2⟩\langle 9, 2 \rangle adds the coordinates, 2+72 + 7 and −1+3-1 + 3. A vector between two points uses their differences.
  • D. Correct. w=⟨7−2,3−(−1)⟩=⟨5,4⟩\mathbf{w} = \langle 7 - 2, 3 - (-1) \rangle = \langle 5, 4 \rangle, also written 5i+4j5\mathbf{i} + 4\mathbf{j}.

Worked example Medium

A hiker walks 10 kilometers east, then 6 kilometers north, then 2 kilometers west, then 9 kilometers north. How many kilometers is the hiker from the starting point?

  1. 1717Answer
  2. 3413\sqrt{41}
  3. 2323
  4. 2727

How to solve it

  1. Write each leg as a vector (east and north positive): ⟨10,0⟩\langle 10, 0 \rangle, ⟨0,6⟩\langle 0, 6 \rangle, ⟨−2,0⟩\langle -2, 0 \rangle, ⟨0,9⟩\langle 0, 9 \rangle.
  2. Add: ⟨10−2,6+9⟩=⟨8,15⟩\langle 10 - 2, 6 + 9 \rangle = \langle 8, 15 \rangle.
  3. Magnitude: 82+152=289=17\sqrt{8^2 + 15^2} = \sqrt{289} = 17.

Why each choice is right or wrong

  • A. Correct. The legs add to ⟨8,15⟩\langle 8, 15 \rangle, and its magnitude is 64+225=289=17\sqrt{64 + 225} = \sqrt{289} = 17 kilometers.
  • B. Incorrect. 3413\sqrt{41} is 369\sqrt{369}, the magnitude of ⟨12,15⟩\langle 12, 15 \rangle, which adds the 2 kilometers west instead of subtracting them. West is the negative direction.
  • C. Incorrect. 23 adds the components of the total, 8+158 + 15. The distance is the magnitude, 82+152\sqrt{8^2 + 15^2}.
  • D. Incorrect. 27 is 10+6+2+910 + 6 + 2 + 9, the total distance walked. The question asks how far the hiker ends up from the start.

Trap Terminal minus initial

The vector from PP to QQ is Q−PQ - P. Subtracting the other way gives a vector of the same length that points the opposite way, and copying a negative coordinate without its sign changes a component. Write the subtraction out, parentheses and all: 3−(−1)3 - (-1).

Trap Distance walked is not distance from the start

Adding the lengths of the legs gives the distance walked, which is longer than the distance from the start unless every leg points the same way. Adding the two components of the total, such as 8+158 + 15, is another trap: the length of ⟨8,15⟩\langle 8, 15 \rangle is 82+152=17\sqrt{8^2 + 15^2} = 17.

Calculator When a calculator isn’t faster

Do vector arithmetic by hand: each component is one subtraction or multiplication. A calculator helps with the square root in a magnitude when the sum of squares is not a perfect square. If you want to see a direction, you can plot the two points in the built-in Desmos calculator if you test online, but that rarely saves time.