Any vector in a plane can be represented as a combination of two nonparallel vectors.
Linear Independence
When two vectors →a, →b satisfy →a ≠ →0, →b ≠ →0, and →a and →b are not parallel, →a and →b are called linearly independent.
Decomposition Using Linearly Independent Vectors
When →a and →b are linearly independent, any vector →p in the plane can be represented uniquely as
→p = s→a + t→b (s and t are real numbers).
Also, comparison of coefficients: s→a + t→b = s'→a + t'→b ⇒ s=s'、t=t'.
Example ①
When →a=(2, 1) and →b=(1, −3), express →p=(5, −4) as →p = s→a + t→b.
Solution: (2s+t, s−3t)=(5, −4). The simultaneous equations are 2s+t=5 and s−3t=−4.
Substitute t=5−2s from the first equation: s−3(5−2s)=−4 → 7s=11 → s=11/7, t=13/7. Therefore, s=11/7, t=13/7.
When →a=(2, 1) and →b=(1, −3), express →p=(5, −4) as →p = s→a + t→b.
Solution: (2s+t, s−3t)=(5, −4). The simultaneous equations are 2s+t=5 and s−3t=−4.
Substitute t=5−2s from the first equation: s−3(5−2s)=−4 → 7s=11 → s=11/7, t=13/7. Therefore, s=11/7, t=13/7.
Use in Geometry Problems
In triangle OAB, let →OA=→a and →OB=→b. Internal division points and midpoints can be represented using →a and →b.
Midpoint M of AB: →OM = (→a + →b)/2
Point P internally dividing AB in the ratio m:n: →OP = (n→a + m→b)/(m+n)
Example ②
In triangle OAB, →OA=→a and →OB=→b. Express →OP for point P that internally divides side AB in the ratio 2:1.
Solution: →OP = (1→a + 2→b)/3 = (1/3)→a + (2/3)→b
In triangle OAB, →OA=→a and →OB=→b. Express →OP for point P that internally divides side AB in the ratio 2:1.
Solution: →OP = (1→a + 2→b)/3 = (1/3)→a + (2/3)→b
Key Points
- Linear independence: two nonzero, nonparallel vectors
- Any vector can be uniquely decomposed as →p = s→a + t→b
- Simultaneous equations can be formed by comparing coefficients
- Internal division formula: division in m:n gives →OP=(n→a+m→b)/(m+n)
Practice Problems
- When →a=(1, 2) and →b=(3, −1), express →p=(5, 0) as s→a + t→b.
- In triangle OAB, →OA=→a and →OB=→b. Let M be the midpoint of AB. Express →OM.
- →a and →b are linearly independent. If s→a + t→b = 3→a − 2→b, find s and t.
Answers and Explanations
- Answer: s=5/7, t=10/7
Explanation: s+3t=5 and 2s−t=0. Substituting t=2s gives 7s=5 → s=5/7, t=10/7. - Answer: →OM=(→a+→b)/2
Explanation: The midpoint vector formula. - Answer: s=3, t=−2
Explanation: Compare coefficients because the vectors are linearly independent.
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