How to Prove Three Vectors or Points Are Collinear in Geometry
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 Understanding Collinear Vectors: Conditions, Formulas & Proofs
1. What are collinear vectors?
Collinear vectors are vectors that lie along the same straight line or on parallel lines, meaning the direction of one vector is either the same as or exactly opposite to the other. In other words, two or more vectors are collinear if they have the same or exactly opposite direction irrespective of their magnitudes.
2. How do you know if vectors are collinear?
Two vectors are collinear if one is a scalar multiple of the other. Mathematically, vectors a and b are collinear if a = k × b, where k is a scalar. Alternatively, their cross product will be zero (for three-dimensional vectors).
3. What is the formula for collinear vectors?
The condition for collinear vectors is that two vectors a = (a1, a2, a3) and b = (b1, b2, b3) are collinear if:
(a1/b1) = (a2/b2) = (a3/b3), provided denominators are not zero.
Alternatively, for 3D vectors, a × b = 0.
4. What is the condition for two vectors to be collinear?
The condition for two vectors to be collinear is that one vector must be a scalar multiple of the other. If vector a = λ × vector b, where λ is a real number (scalar), the vectors are collinear. Also, the cross product a × b = 0 for collinear vectors.
5. How to prove that three points are collinear?
To prove that three points (A, B, C) are collinear, check if the vectors AB and AC form a straight line:
1. Find AB = B - A and AC = C - A
2. The points are collinear if AB and AC are collinear vectors (i.e., AB = k × AC for some scalar k), or if the area of triangle ABC is zero.
6. How do you visually represent collinear vectors?
On a diagram, collinear vectors are shown as arrows lying along the same straight line, either pointing in the same or exactly opposite directions with respect to a common origin or anywhere on that line. They do not form angles other than 0° or 180° between each other.
7. What is the difference between collinear vectors and parallel vectors?
While both collinear and parallel vectors point in the same or opposite directions, collinear vectors must lie on the same line (or be scalar multiples of each other). Parallel vectors can lie on different lines but maintain a constant angle (0° or 180°) with each other. All collinear vectors are parallel, but not all parallel vectors are strictly collinear.
8. What are the important properties of collinear vectors?
Properties of collinear vectors include:
• They have the same or opposite direction.
• They can be represented as scalar multiples of each other.
• The angle between them is 0° or 180°.
• Their cross product equals zero.
• They are linearly dependent.
9. What are non-collinear vectors?
**Non-collinear vectors** are vectors that do not lie on the same straight line and cannot be expressed as scalar multiples of each other. The angle between non-collinear vectors is not 0° or 180°, and their cross product is not zero. Three non-collinear points or vectors form a triangle and do not lie on the same straight line.
10. What is an example of collinear vectors?
For example, let vector a = (2, 4, 6) and vector b = (1, 2, 3). Here, a = 2 × b, so both are collinear vectors since one is a scalar multiple of the other.
11. How are collinear vectors used in physics?
In physics, collinear vectors often describe forces acting along the same straight line, like tension in a rope or opposite electric charges on a single axis. Identifying collinear vectors helps in analyzing equilibrium and net force calculations.
12. How do you solve questions on collinear vectors for exams?
To solve collinear vectors questions in exams:
• Check if vectors are scalar multiples.
• Verify the cross product (for 3D vectors) is zero.
• For points, ensure the area of the formed triangle is zero.
• Substitute given values and solve for the unknown scalar if required.
