Arrows you can add
Helpful first: Pythagoras and square roots, for lengths. Ready if you can find the long side of a right triangle with short sides 6 and 8.
Tonight's idea
Some things need two pieces of information. "Walk 5 km" is not enough; you need to know 5 km which way. A vector is an arrow: it has a length and a direction, and nothing else. Slide it anywhere on the page and it is still the same vector.
The easiest way to write a vector is as steps on a grid, like coordinates (see Below zero): \((3, 2)\) means 3 steps east and 2 steps north. \((-1, 4)\) means 1 step west and 4 north.
Two rules do almost everything.
Adding. Put the tail of the second arrow on the head of the first. The sum is the arrow from the very start to the very end. In coordinates you just add the parts: \((3, 2) + (-1, 4) = (2, 6)\).
Scaling. Multiply a vector by a number and it stretches: \(2 \times (3, 2) = (6, 4)\), same direction, twice as long. Multiply by \(-1\) and it points the opposite way.
The length of a vector comes from Pythagoras: \((3, 4)\) has length \(\sqrt{3^2 + 4^2} = 5\).
Physics uses vectors for anything with a direction: velocity, force, wind, gravity. A matrix turns one vector into another.
Watch one: the river
A boat points straight across a river and moves at 3 m/s through the water. The current carries everything downstream at 4 m/s. Where does the boat actually go?
Two velocities (speeds with a direction) act at once, so add them. Across is \((0, 3)\) and downstream is \((4, 0)\). The sum is \((4, 3)\): every second the boat ends up 4 m downstream and 3 m across. Its real speed is the length of that arrow, \(\sqrt{4^2 + 3^2} = 5\) m/s, and its path is a straight diagonal line, not the straight-across line the boat is pointing along.
If the river is 30 m wide, crossing takes \(30 / 3 = 10\) seconds (only the across part matters for that), and in those 10 seconds the current carries the boat \(4 \times 10 = 40\) m downstream.
Your turn
Tonight's core: the example and exercises 1 to 3. The rest can wait for another visit.
- Let \(\mathbf{a} = (4, 1)\) and \(\mathbf{b} = (-2, 3)\). Find \(\mathbf{a} + \mathbf{b}\), \(\mathbf{a} - \mathbf{b}\), and \(3\mathbf{a}\). Draw \(\mathbf{a} + \mathbf{b}\) head to tail on squared paper and check it matches.
- Find the length of \((6, 8)\). Then find the length of \((-6, 8)\) without recomputing. Why is it the same?
- A swimmer heads straight across a 60 m river at 2 m/s. The current is 1.5 m/s. Write both velocities as vectors, add them, find the swimmer's actual speed, and work out how far downstream they land.
- Treasure map. From the old oak, walk 3 paces east and 4 north, then 2 west and 1 north. Write the two legs as vectors, add them, and say how far you are from the oak in a straight line.
- A robot does these moves in order: \((2, 0)\), \((0, 3)\), \((-2, 0)\), and one more move you must find so that it ends back where it started. What is the missing vector, and what is the total of all four?
- Describe in words what \(-2 \times (3, 1)\) looks like compared with \((3, 1)\). Then give its coordinates.
Stretch: why order does not matter
Draw \(\mathbf{a}\) then \(\mathbf{b}\) head to tail. Now, from the same starting point, draw \(\mathbf{b}\) then \(\mathbf{a}\). The two routes form a four-sided shape. What kind of shape is it, and why does that prove \(\mathbf{a} + \mathbf{b} = \mathbf{b} + \mathbf{a}\) for every pair of vectors?
Then: points \(P = (2, 6)\) and \(Q = (8, 2)\). The vector from \(P\) to \(Q\) is \(Q - P\) (think of each point as the arrow from \((0,0)\) to it). Half of that, added to \(P\), lands you where? Compare with the average of the two points, \(\left(\frac{2+8}{2}, \frac{6+2}{2}\right)\).
Answers
- \(\mathbf{a} + \mathbf{b} = (2, 4)\). \(\mathbf{a} - \mathbf{b} = (6, -2)\). \(3\mathbf{a} = (12, 3)\).
- \(\sqrt{36 + 64} = \sqrt{100} = 10\). \((-6, 8)\) also has length 10: squaring removes the minus sign, and geometrically it is the mirror image, same length.
- Across \((0, 2)\), current \((1.5, 0)\), sum \((1.5, 2)\). Speed \(\sqrt{1.5^2 + 2^2} = \sqrt{6.25} = 2.5\) m/s. Crossing takes \(60 / 2 = 30\) s, so the swimmer lands \(1.5 \times 30 = 45\) m downstream.
- \((3, 4) + (-2, 1) = (1, 5)\). Distance \(\sqrt{1 + 25} = \sqrt{26} \approx 5.1\) paces, even though you took 10 paces in total (the first leg alone covers 5 paces as the crow flies).
- So far: \((2, 0) + (0, 3) + (-2, 0) = (0, 3)\). The missing move is \((0, -3)\). The total of all four is \((0, 0)\), the zero vector: any closed loop of moves adds to zero.
- Same line, opposite direction, twice as long: \((-6, -2)\).
Stretch. The two routes make a parallelogram, because opposite sides are the same vector (same length, same direction). Both routes end at the opposite corner, so the two sums are the same arrow. For the midpoint: \(Q - P = (6, -4)\), half is \((3, -2)\), and \(P + (3, -2) = (5, 4)\). The average of the points is also \((5, 4)\). The midpoint is the average of the position vectors, always.