Vectors

Question 1

If a line (AB) is split by a point (P) in a ratio, how to find AP and PB if ratio is 2:3

Answer 1

AP = 2/5 AP
PB = 3/5 AP

Question 2

Express OP in terms of a and c (given a shape e.g square)

Answer 2

1. Write down all info e.g squares sides are equal so AB = OC
2. Find 2 expressions for the thing u wanna find
3. Set equal to each other and solve

Question 3

Show P divides AC in ratio 2:3

Answer 3

1. Find expression for AP and PB
2. Simplify and should get fraction(same bracket)

Question 4

Parallel to ai + bj

Answer 4

1. Use fractions as the ratios are the same
2. Do ..i + ..j = K(ai + bj) and solve for K

Question 5

What is the scalar/dot product?

Answer 5

(i1 x i2) + (j1 x j2) + (k1 x k2)

Question 6

How to find the unit vector

Answer 6

a / |a|

Question 7

What does unit vectors length =

Answer 7

1

Question 8

How to find AB

Answer 8

OB - OA

Question 9

How to find angles between 2 lines

Answer 9

Cosθ = (a · b) / (|a| |b|)

Question 10

When finding angle BAC what 2 lines would u use

Answer 10

BA and BC
MUST BE FROM SAME STARTING POINT

Question 11

Find dot product a · b

Answer 11

(X1 * X2) + (Y1 * Y2) + (Z1 * Z2)

Question 12

If θ is acute?

Answer 12

Cos θ > 0

Question 13

If θ is obtuse

Answer 13

Cos θ < 0

Question 14

Area of triangle with angle

Answer 14

Area = ab * sin C

Question 15

Area parallelogram with angle

Answer 15

Area = ab * sinC

Question 16

Area of trapezium with angle

Answer 16

1) Split into a rectangle and 2 triangle

Question 17

Show 2 lines are parallel

Answer 17

USE DIRECTION VECTORS ONLY (d1 and d2)
1) d1 = m*d2

Question 18

Show 2 lines perpendicular

Answer 18

USE DIRECTION VECTOR ONLY
d1 • d2 = 0

Question 19

2 lines intersect

Answer 19

1. Vector equations equal each other
2. Use 2 in simultaneous equations
3. Solve for 人 and U
4. If those values work for the last equations = intersection at those values

Question 20

Skew means

Answer 20

Aren’t parallel and don’t intersect

Question 21

Prove for all values of b≠a these lines are skew

Answer 21

1. Prove aren’t parallel
2. As do intersection show only way wld intersect is if b=a
3. Therefore if b≠ a then are skew

Question 22

Point on the line closest to (origin or another point)

Answer 22

Means perpendicular to that point
E.g X closest to origin = X is perpendicular to origin
Thus dot product of X and Origin = 0

Question 23

B is the image of A reflected in L, find B

Answer 23

AX = XB
XB = B - X
So know what AX is from earlier/find it
… = a/b/c - (x value)
Solve for a/b/c

Question 24

Find the possible positions of R given PQ=PR

Answer 24

1. Length of PQ and PR are same
2. Vector equation of R
3. Put into distance/length formula and set equal to whatever PQ length is
4. When square rooting, answer is +- so have 2 values u can use

Question 25

Find the 2 positions of D given the area of triangles CAD is twice the area of CAB

Answer 25

1. One Height or length will be same 2. Find length of AB and direction vector 3. AD is 2X AB 4. How much is AD x the direction vector? 5. Substitute this value +ve and -ve into the line equations as all on same line

Question 26

Angle between 2 lines must be from e.g use AB and BC?

Answer 26

Same starting point