The vector m is oppositely directed to the vector b {-2; 4} and has the length of the vector a {2; 2}.

The vector m is oppositely directed to the vector b {-2; 4} and has the length of the vector a {2; 2}. Find the coordinates of the vector m.

1. Since the vector m is oppositely directed to the vector b (- 2; 4), the coordinates of the vector m will be (- 2 * (- x); 4 * (- x)) = (2 * x; – 4 * x) …
2. The length of the vector a with coordinates (2; 2) will be equal to:
| a | = √ (2² + 2²) = √ (4 + 4) = √ (2 * 4) = 2√2.
3. The length of the vector m with coordinates (- 2 * x; 4 * x) will be equal to:
| m | = √ ((2 * x) ² + (- 4 * x) ²) = √ (4 * x² + 16 * x²) = √ (20 * x²) = 2 * x√5.
4. Since the length of the vector m is equal to the length of the vector a, then:
| a | = | m |;
2 * x√5 = 2√2;
x = (2√2) / (2√5) (proportional);
x = √2 / √5;
x = √10 / 5.
5. The coordinates of the vector m are:
m = (2 * x; – 4 * x) = (2√10 / 5; – 4√10 / 5).
Answer: m (2√10 / 5; – 4√10 / 5).