How do you know if two vectors are parallel or perpendicular?
Understanding parallel vectors is essential for solving geometry, physics, and vector algebra questions in school board exams and engineering entrance tests. Knowing when vectors are in the same or opposite direction makes it easier to tackle coordinate geometry, 3D problems, and real-life direction-based questions confidently.
Formula Used in Parallel Vectors
The standard formula is: \( \vec{a} = k\vec{b} \) — where two vectors \( \vec{a} \) and \( \vec{b} \) are parallel if one is a scalar multiple of the other. Alternatively, for two vectors \( \vec{a} \) and \( \vec{b} \), they are parallel if \( \vec{a} \times \vec{b} = 0 \).
Here’s a helpful table to understand parallel vectors more clearly:
Parallel Vectors Table
| Vectors | Are They Parallel? | Why/Why Not |
|---|---|---|
| (2, 4), (4, 8) | Yes | Second is 2× the first (scalar multiple) |
| (3, 2), (6, -4) | No | Direction ratios not proportional |
| (1, -1, 2), (2, -2, 4) | Yes | All components proportional (×2) |
| (2, 5), (5, 2) | No | Components not in same ratio |
This table shows how recognising proportional components helps to identify parallel vectors quickly in various problems.
Worked Example – Solving a Problem
1. Determine if the vectors \( \vec{a} = (10, -6) \) and \( \vec{b} = (15, -9) \) are parallel.
2. Find a scalar \( k \) such that \( \vec{a} = k\vec{b} \).
\( -6 = -9k \implies k = \frac{2}{3} \)
3. Since \( k \) is the same for both components, \( \vec{a} \) and \( \vec{b} \) are parallel.
Final Answer: Yes, the vectors are parallel because their components are proportional.
Practice Problems
- Check if the vectors (3, 6) and (1, 2) are parallel.
- Find a unit vector parallel to (6, 8).
- Give an example of two anti-parallel vectors in 3D.
- State if (2, 5, -7) and (4, 10, -14) are parallel.
Common Mistakes to Avoid
- Confusing parallel vectors with equal vectors; remember, magnitudes can differ.
- Not checking every component for proportionality in 3D vectors.
- Thinking parallel always means “same direction” — opposite direction vectors can be parallel too.
Real-World Applications
The concept of parallel vectors is used in navigation, construction, and engineering — for example, ensuring beams and force directions remain aligned when building bridges or designing maps. Vedantu lessons connect these abstract ideas to real-world scenarios, helping you see maths in action.
We explored the idea of parallel vectors, how to test for them using ratios or cross products, solved examples step by step, and saw their practical uses. Practice more problems and use interactive resources at Vedantu to achieve mastery with parallel vectors.
Want to deepen your understanding of vector relationships and operations? Visit Vector Algebra for more detailed explanations, or explore Vector Cross Product to learn how the cross product test confirms when vectors are parallel. For geometric insights, see Lines Parallel to the Same Line and Vector Direction and Ratios to strengthen your understanding of direction cosines and ratios.
FAQs on What Are Parallel Vectors? Complete Guide for Students
1. How do you know if two vectors are parallel?
Two vectors are parallel if they point in exactly the same direction or exactly opposite directions. Mathematically, vectors a and b are parallel if one is a scalar multiple of the other: a = k × b where k is a non-zero constant. This means their ratios of corresponding components are equal.
2. What is the formula for checking if two vectors are parallel or perpendicular?
To check if two vectors are parallel, verify if a = k × b for some scalar k ≠ 0.
To check if they are perpendicular, calculate their dot product; if a • b = 0, the vectors are perpendicular or orthogonal to each other.
3. What is the difference between a like vector and a parallel vector?
A like vector has the same direction and the same sense (points the same way) as another vector, but may differ in magnitude. Parallel vectors have the same or exactly opposite direction, regardless of magnitude or sense. So, all like vectors are parallel, but not all parallel vectors are like vectors.
4. What is the definition of two parallel vectors?
Two vectors are called parallel if they have the same or exactly opposite direction. Algebraically, a and b are parallel if a = k × b for some non-zero scalar k.
5. What is the dot product of two parallel vectors?
The dot product of two parallel vectors is equal to the product of their magnitudes, i.e., a • b = |a| × |b|, if they point in the same direction. If they point in opposite directions, a • b = -|a| × |b|.
6. If two vectors are parallel, do they have the same magnitude?
No, parallel vectors do not necessarily have the same magnitude. They only need to have the same or opposite direction. Magnitude may be different unless specified.
7. What is the cross product of parallel vectors?
The cross product of two parallel vectors is always zero because the sine of the angle between them is zero (θ = 0° or 180°). So, a × b = 0 for parallel vectors.
8. How do you find if vectors are parallel in 3D?
In 3D, two vectors are parallel if their components are proportional. For vectors a = (a1, a2, a3) and b = (b1, b2, b3), if a1/b1 = a2/b2 = a3/b3, then they are parallel (provided none of the denominators are zero).
9. What are the properties of parallel vectors?
The main properties of parallel vectors are:
• Their direction vectors are scalar multiples.
• The angle between them is 0° or 180°.
• Their cross product is zero.
• They may have different magnitudes.
• They can have the same or opposite sense.
10. What does it mean when vectors are collinear?
Two or more vectors are collinear if they lie along the same straight line, which means they are also parallel. Collinear vectors must be parallel, but parallel vectors do not always have to be collinear if they originate from different lines.
11. Can you give an example of parallel vectors?
Yes. For example, vector a = (2, 4, 6) and vector b = (1, 2, 3) are parallel because a = 2 × b. Their components are in the same proportion.
12. What is the condition for two vectors to be parallel?
The condition for two vectors a and b to be parallel is: a = k × b for some scalar k ≠ 0. This applies in 2D and 3D. Alternatively, their cross product is zero.
