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Vector geometry

Write a vector with a and b · Precalculus · 20 problems, easy to harder

▶ Watch the first one solved

How we do it

  1. Find a route
  2. Write it with a and b
  3. Collect the a and b
  4. Check

The steps for OA = a, OB = b, find AB. Every solution names its steps, so you always know what comes next.

Worked examples

OA = a, OB = b, find AB Easy

The question

Question: O A is the vector a and O B is the vector b: find the vector A B.

1 Find a route

We want to go from A to B.

We go along arrows we know: A O, then O B.

So A B is A O plus O B.

AB = AO + OB

2 Write it with a and b

Careful!

Going against the arrow is the opposite vector: it gets a minus.

Careful: not AO = a ✗

A O goes against the arrow a: minus a.

O B is b.

AB = −a + b

3 Collect the a and b

The a's together and the b's together: b minus a.

AB = b − a
a terms + a terms, b terms + b terms

4 Check

Let's check with numbers: a = (0, 4) and b = (6, 0).

Our answer gives (6, -4): exactly the step from A to B.

So it is right!

a = (0, 4), b = (6, 0): AB = (6, -4) = B(6, 0) − A(0, 4)

Let's look back.

First: find a route.

Then: write it with a and b.

Then: collect the a and b.

Last: check.

Answer
a terms + a terms, b terms + b terms

▶ Watch it solved

OA = a, OB = b, M is the midpoint of AB, find OM Medium

The question

Question: O A is the vector a and O B is the vector b, and M is the midpoint of A B: find the vector O M.

1 Find a route

We want to go from O to M.

We go along arrows we know: O A, then A M.

So O M is O A plus A M.

OM = OA + AM

2 Write it with a and b

Careful!

A B goes from A to B: b minus a, not a minus b.

Careful: not AM = ½(a − b) ✗

O A is a.

A M is half of A B, and A B is b minus a.

OM = a + (12b − 12a)

3 Collect the a and b

The a's together and the b's together: a half a plus a half b.

OM = 12a + 12b
a terms + a terms, b terms + b terms

4 Check

Let's check with numbers: a = (0, 4) and b = (6, 0).

Our answer gives (3, 2): exactly the step from O to M.

So it is right!

a = (0, 4), b = (6, 0): OM = (3, 2) = M(3, 2) − O(0, 0)

Let's look back.

First: find a route.

Then: write it with a and b.

Then: collect the a and b.

Last: check.

Answer
a terms + a terms, b terms + b terms

▶ Watch it solved

OA = a, OB = b, P divides AB in the ratio 2 : 3, find OP Harder

The question

Question: O A is the vector a and O B is the vector b, and P divides A B in the ratio 2 : 3: find the vector O P.

1 Find a route

We want to go from O to P.

We go along arrows we know: O A, then A P.

So O P is O A plus A P.

OP = OA + AP

2 Write it with a and b

Careful!

A B goes from A to B: b minus a, not a minus b.

Careful: not AP = (a − b) ✗

O A is a.

A P is 2 of the 5 parts of A B, and A B is b minus a.

OP = a + (25b − 25a)

3 Collect the a and b

The a's together and the b's together: three fifths a plus two fifths b.

OP = 35a + 25b
a terms + a terms, b terms + b terms

4 Check

Let's check with numbers: a = (0, 4) and b = (6, 0).

Our answer gives (2.4, 2.4): exactly the step from O to P.

So it is right!

a = (0, 4), b = (6, 0): OP = (2.4, 2.4) = P(2.4, 2.4) − O(0, 0)

Let's look back.

First: find a route.

Then: write it with a and b.

Then: collect the a and b.

Last: check.

Answer
a terms + a terms, b terms + b terms

▶ Watch it solved

Practice

Try each one, then open the answer. Stuck? Open the problem to watch every step.

Easy

  1. OA = a, OB = b, find AB
    Answer
    a terms + a terms, b terms + b terms
  2. OA = a, OB = b, find BA
    Answer
    a terms + a terms, b terms + b terms
  3. OA = a, OB = b, N is the midpoint of OB, find ON
    Answer
    ON = 12b
  4. OA = a, OB = b, K is the midpoint of OA, find OK
    Answer
    OK = 12a
  5. OA = a, OB = b, N is the midpoint of OB, find AN
    Answer
    a terms + a terms, b terms + b terms
  6. OA = a, OB = b, K is the midpoint of OA, find BK
    Answer
    a terms + a terms, b terms + b terms

Medium

  1. OA = a, OB = b, M is the midpoint of AB, find OM
    Answer
    a terms + a terms, b terms + b terms
  2. OA = a, OB = b, M is the midpoint of AB, find AM
    Answer
    a terms + a terms, b terms + b terms
  3. OA = a, OB = b, M is the midpoint of AB, find BM
    Answer
    a terms + a terms, b terms + b terms
  4. OA = a, OB = b, P divides AB in the ratio 1 : 2, find OP
    Answer
    a terms + a terms, b terms + b terms
  5. OA = a, OB = b, P divides AB in the ratio 2 : 1, find OP
    Answer
    a terms + a terms, b terms + b terms
  6. OA = a, OB = b, P divides AB in the ratio 1 : 3, find OP
    Answer
    a terms + a terms, b terms + b terms
  7. OA = a, OB = b, P divides AB in the ratio 3 : 1, find OP
    Answer
    a terms + a terms, b terms + b terms

Harder

  1. OA = a, OB = b, P divides AB in the ratio 2 : 3, find OP
    Answer
    a terms + a terms, b terms + b terms
  2. OA = a, OB = b, P divides AB in the ratio 1 : 2, find AP
    Answer
    a terms + a terms, b terms + b terms
  3. OA = a, OB = b, P divides AB in the ratio 3 : 2, find BP
    Answer
    a terms + a terms, b terms + b terms
  4. OA = a, OB = b, P divides AB in the ratio 1 : 4, find OP
    Answer
    a terms + a terms, b terms + b terms
  5. OA = a, OB = b, P divides AB in the ratio 3 : 1, find AP
    Answer
    a terms + a terms, b terms + b terms
  6. OA = a, OB = b, P divides AB in the ratio 2 : 1, find BP
    Answer
    a terms + a terms, b terms + b terms
  7. OA = a, OB = b, P divides AB in the ratio 4 : 1, find OP
    Answer
    a terms + a terms, b terms + b terms