Geometry and measures, Year 11: vectors

2; 3

HintAdd the two top numbers together, and separately the two bottom numbers.

The whyTop: 3 + (−1) = 2. Bottom: −2 + 5 = 3. Adding vectors means doing one journey and then the other: 3 right and 2 down, then 1 left and 5 up, is 2 right and 3 up overall.

Top 2, bottom −5

HintDouble every number in b first, then subtract row by row.

The why2b has 2 on top and 6 underneath. Subtracting: top 4 − 2 = 2, bottom 1 − 6 = −5. Multiplying a vector by a number multiplies both of its components.

−a

HintIt is the same journey done backwards.

The whyGoing from B to A covers the same distance in the opposite direction, and reversing a vector changes its sign. In a column vector both numbers change sign.

a + b

HintGo from A to C by way of B.

The whyA vector describes a movement, and any route between the same two points gives the same overall movement. Going from A to B and then from B to C is a followed by b, which is a + b.

A scalar

HintIt is a quantity with size but no direction; its name is linked to making things bigger or smaller.

The whyMultiplying a vector by a scalar changes its length but keeps it parallel: 3a is three times as long as a and points the same way. A negative scalar reverses the direction as well.

b − a

HintTravel from A back to O, then on from O to B.

The whyFrom A to O is −a and from O to B is b, so from A to B is −a + b = b − a. This 'end minus start' result is the first step in most vector proofs.

3/4 a + 1/4 b

HintP is a fraction of the way along AB. Go from O to A, then add that fraction of the vector from A to B.

The whyThe ratio 1 : 3 makes AP one quarter of AB. From A to B is b − a, so from O to P is a + 1/4 (b − a) = 3/4 a + 1/4 b. P is nearer to A, so a has the bigger share.

They are parallel, and RS is 1.5 times as long as PQ

HintTake a common factor out of each vector and compare what is left in the brackets.

The why2a + 4b = 2(a + 2b) and 3a + 6b = 3(a + 2b). Both are multiples of a + 2b, so the vector from R to S is 3/2 of the vector from P to Q. One vector being a multiple of another proves the lines are parallel and gives the ratio of their lengths.

The vector from B to C is 3 times the vector from A to B, so BC is parallel to AB, and the two lines share the point B

HintCompare the two journeys by factorising, then notice where one ends and the next begins.

The why3a + 6b = 3(a + 2b), so the directions are the same. Two parallel lines that pass through a common point must be the same line, so A, B and C are collinear, with BC three times as long as AB.

M to N is b − a and A to B is 2b − 2a, so MN is parallel to AB and half as long

HintReach each end point by going back through O.

The whyFrom M to N: −a + b = b − a. From A to B: −2a + 2b = 2(b − a). The second is twice the first, so the lines are parallel and AB is twice as long as MN. The same is true for the line joining the midpoints of two sides of any triangle.