Geometry and measures, Year 11: vectors

2; 3

提示Add the two top numbers together, and separately the two bottom numbers.

为什么Top: 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

提示Double every number in b first, then subtract row by row.

为什么2b 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

提示It is the same journey done backwards.

为什么Going 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

提示Go from A to C by way of B.

为什么A 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

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

为什么Multiplying 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

提示Travel from A back to O, then on from O to B.

为什么From 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

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

为什么The 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

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

为什么2a + 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

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

为什么3a + 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

提示Reach each end point by going back through O.

为什么From 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.