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What is the Vector Equation of a Line Passing Through a Point and Parallel to a Vector?
Grade Level:
Class 12
AI/ML, Physics, Biotechnology, FinTech, EVs, Space Technology, Climate Science, Blockchain, Medicine, Engineering, Law, Economics
Definition
What is it?
The vector equation of a line passing through a specific point and parallel to a given vector helps us describe the line's path using vectors. It tells us the position of any point on the line relative to the origin, using a starting point and a direction.
Simple Example
Quick Example
Imagine you are at your friend's house (a specific point) and you want to walk straight to the market. You know the direction the market is in (parallel to a vector). The vector equation is like giving directions to anyone from your friend's house to any shop on that straight road to the market.
Worked Example
Step-by-Step
PROBLEM: Find the vector equation of a line passing through the point A (1, 2, 3) and parallel to the vector b = i + 2j - 3k.
STEP 1: Identify the position vector of the given point. The point A (1, 2, 3) has a position vector a = i + 2j + 3k.
---STEP 2: Identify the parallel vector. The line is parallel to vector b = i + 2j - 3k.
---STEP 3: Recall the general form of the vector equation of a line. It is given by r = a + λb, where 'r' is the position vector of any point on the line, 'a' is the position vector of the known point, 'b' is the parallel vector, and 'λ' (lambda) is a scalar parameter.
---STEP 4: Substitute the identified vectors into the general form. r = (i + 2j + 3k) + λ(i + 2j - 3k).
ANSWER: The vector equation of the line is r = (i + 2j + 3k) + λ(i + 2j - 3k).
Why It Matters
Understanding vector equations helps engineers design roads and robot paths in AI/ML. In Physics, it's crucial for tracking satellite movements or missile trajectories for ISRO. Even in FinTech, complex algorithms use similar logic to model data trends, showing how mathematics helps build our future.
Common Mistakes
MISTAKE: Confusing the given point's coordinates with the parallel vector's components. | CORRECTION: The point's coordinates form the 'a' vector (position vector), while the direction components form the 'b' vector (direction vector).
MISTAKE: Forgetting the scalar parameter (lambda, λ) in the equation. | CORRECTION: The 'λ' is essential because it allows 'b' to be scaled, representing all points along the line, not just one specific point in the direction of 'b'.
MISTAKE: Writing the equation as r = b + λa. | CORRECTION: The correct form is r = a + λb. 'a' is the starting point, and 'λb' is the displacement from that starting point along the line.
Practice Questions
Try It Yourself
QUESTION: Find the vector equation of a line passing through the point P (2, -1, 4) and parallel to the vector d = 3i + j - 2k. | ANSWER: r = (2i - j + 4k) + λ(3i + j - 2k)
QUESTION: A line passes through the origin (0,0,0) and is parallel to the vector v = -i + 5j + k. Write its vector equation. | ANSWER: r = λ(-i + 5j + k)
QUESTION: If a line passes through point A (1,1,1) and is parallel to the line whose equation is r = (2i - j + 3k) + μ(4i + 2j - k), find the vector equation of the first line. | ANSWER: r = (i + j + k) + λ(4i + 2j - k)
MCQ
Quick Quiz
Which of the following is the correct vector equation of a line passing through point (x1, y1, z1) and parallel to vector ai + bj + ck?
r = (x1i + y1j + z1k) + (ai + bj + ck)
r = λ(x1i + y1j + z1k) + (ai + bj + ck)
r = (x1i + y1j + z1k) + λ(ai + bj + ck)
r = (ai + bj + ck) + λ(x1i + y1j + z1k)
The Correct Answer Is:
C
The correct form is r = a + λb, where 'a' is the position vector of the point and 'b' is the parallel vector. Option C correctly places the position vector of the point first, followed by the scalar multiple of the parallel vector.
Real World Connection
In the Real World
Imagine a drone delivering a package for Zepto. The drone takes off from a specific location (a point) and needs to fly in a straight path towards the customer. The vector equation helps calculate this straight path, ensuring the drone follows the correct direction and reaches its destination efficiently.
Key Vocabulary
Key Terms
VECTOR: A quantity having both magnitude and direction | POSITION VECTOR: A vector representing the position of a point relative to the origin | PARALLEL VECTOR: A vector that gives the direction of the line | SCALAR PARAMETER (λ): A variable that scales the direction vector to cover all points on the line
What's Next
What to Learn Next
Next, you can learn about the Cartesian equation of a line, which is another way to represent the same line using coordinates. This will help you switch between different forms and solve more complex problems in 3D geometry.
