Collinear Vectors Definition Formula and Solved Examples
Collinear Vectors are essential for solving many Class 11 and 12 Maths and Physics questions, especially when checking whether points or forces lie in a straight line. Learning this concept makes it easier to approach exam problems involving vector direction, parallelism, and geometry confidently.
Formula Used in Collinear Vectors
The standard formula is: \( \vec{a} = k\vec{b} \), where “k” is a scalar. Alternatively, for vectors \( \vec{a} = (a_1,a_2,a_3) \) and \( \vec{b} = (b_1,b_2,b_3) \), check collinearity with \( \frac{a_1}{b_1} = \frac{a_2}{b_2} = \frac{a_3}{b_3} \) (as long as none are zero).
Here’s a helpful table to understand Collinear Vectors more clearly:
Collinear Vectors Table
| Condition | Description | Applies To |
|---|---|---|
| Scalar Multiple | \( \vec{a} = k\vec{b} \) | All Vectors |
| Equal Ratio of Components | \( \frac{a_1}{b_1} = \frac{a_2}{b_2} = \frac{a_3}{b_3} \) | 3D Vectors |
| Zero Cross Product | \( \vec{a} \times \vec{b} = \vec{0} \) | 3D Vectors |
This table shows the main ways to check for collinear vectors in real problems, especially in board exams or when solving geometry questions.
Worked Example – Solving a Problem
Let’s check if the vectors \( \vec{P}=(3,4,5) \) and \( \vec{Q}=(6,8,10) \) are collinear:
1. Write the vectors clearly: \( \vec{P}=(3,4,5) \), \( \vec{Q}=(6,8,10) \ )2. Find the ratios of each corresponding component:
\( \frac{4}{8} = \frac{1}{2} \)
\( \frac{5}{10} = \frac{1}{2} \)
3. Since all ratios are equal, vectors \( \vec{P} \) and \( \vec{Q} \) are collinear.
Alternatively, you can use the cross product test for 3D vectors. For more on cross products, see Vector Cross Product on Vedantu.
Practice Problems
- Are the vectors \( \vec{a} = (2, 6, -4) \) and \( \vec{b} = (1, 3, -2) \) collinear?
- Show that the vectors joining the points A(1,2), B(3,6), and C(5,10) are collinear.
- If \( \vec{m} = (k, 8, 12) \) and \( \vec{n} = (3, 12, 18) \) are collinear, find the value of k.
- Which of the following are not collinear vectors: \( (1,2,3), (2,4,6), (5,7,8) \)?
Common Mistakes to Avoid
- Confusing collinear vectors with coplanar or just parallel vectors. (For the difference, refer to Coplanar Vectors and Parallel Lines.)
- Using the ratio method when any vector component is zero, which can cause division errors.
Real-World Applications
Checking for collinear vectors is useful in analysing forces in engineering, alignment of objects in physics, and determining if points lie on a straight road or path in navigation. For deeper geometric problems, try studying Vector Equations or Vector Joining Two Points on Vedantu.
We explored the idea of Collinear Vectors, their formulae, stepwise problem-solving, and real uses. Keep practising to master these checks, and visit Vedantu’s Vector Algebra for more detailed explanations and related vector concepts.
FAQs on Collinear Vectors in Vector Algebra
1. What are collinear vectors?
Two or more collinear vectors are vectors that lie along the same straight line or are parallel to the same line. This means one vector is a scalar multiple of the other.
- If \(\vec{a} = k\vec{b}\) for some scalar k, then \(\vec{a}\) and \(\vec{b}\) are collinear.
- They may point in the same or opposite directions.
- Collinearity applies in both 2D and 3D vector geometry.
2. How do you check if two vectors are collinear?
Two vectors are collinear if one is a scalar multiple of the other.
- Let \(\vec{a} = (a_1, a_2)\) and \(\vec{b} = (b_1, b_2)\).
- Check if \(a_1/b_1 = a_2/b_2\) (ratios must be equal).
- In 3D, check if \(a_1/b_1 = a_2/b_2 = a_3/b_3\).
3. What is the condition for collinearity of three points?
Three points are collinear if the vectors formed between them are scalar multiples of each other.
- For points A, B, C, compute vectors \(\vec{AB}\) and \(\vec{AC}\).
- If \(\vec{AB} = k\vec{AC}\), the points lie on the same straight line.
- In coordinate geometry, the area of triangle ABC must be 0.
4. What is the formula for collinear vectors?
The formula for collinear vectors is \(\vec{a} = k\vec{b}\), where k is a scalar.
- In 2D: \(a_1b_2 - a_2b_1 = 0\).
- In 3D: \(\vec{a} \times \vec{b} = \vec{0}\) (cross product equals zero vector).
5. Can you give an example of collinear vectors?
An example of collinear vectors is (3, 6) and (1, 2) because one is a scalar multiple of the other.
- Let \(\vec{a} = (3,6)\) and \(\vec{b} = (1,2)\).
- 3/1 = 3 and 6/2 = 3.
- So, \(\vec{a} = 3\vec{b}\).
6. What is the difference between collinear and parallel vectors?
All collinear vectors are parallel, but parallel vectors may not lie on the same line.
- Collinear vectors: Lie on the same straight line.
- Parallel vectors: Have the same or opposite direction but may be on different lines.
7. How do you prove vectors are collinear using the cross product?
Vectors are collinear in 3D if their cross product equals the zero vector.
- Compute \(\vec{a} \times \vec{b}\).
- If the result is \(\vec{0}\), the vectors are parallel.
- Parallel vectors lying along the same direction are collinear.
8. Why is the area of a triangle zero for collinear points?
The area of a triangle formed by three collinear points is zero because they lie on a straight line.
- Area formula: \(\frac{1}{2}|\vec{AB} \times \vec{AC}|\).
- If points are collinear, \(\vec{AB} \times \vec{AC} = 0\).
- Therefore, area = 0.
9. Are opposite direction vectors collinear?
Yes, vectors in opposite directions are collinear if one is a negative scalar multiple of the other.
- If \(\vec{a} = -k\vec{b}\), where k > 0, they point in opposite directions.
- They still lie on the same straight line.
10. How are collinear vectors used in coordinate geometry?
In coordinate geometry, collinear vectors are used to verify if points lie on the same straight line.
- Check equal slopes between pairs of points.
- Use vector condition \(\vec{AB} = k\vec{AC}\).
- Confirm triangle area equals 0.
