Problem 66 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

If the position vectors of the vertices \(A, B\) and \(C\) of a \(\triangle \mathrm{ABC}\) are respectively \(4 \hat{i}+7 \hat{j}+8 \hat{k}, 2 \hat{i}+3 \hat{j}+4 \hat{k}\) and \(2 \hat{i}+5 \hat{j}+7 \hat{k}\), then the position vector of the point, where 73 the bisector of \(\angle A\) meets \(B C\) is \([\) Online April \(\mathbf{1 5}\), 2018] (a) \(\frac{1}{2}(4 \hat{i}+8 \hat{j}+11 \hat{k})\) (b) \(\frac{1}{3}(6 \hat{i}+13 \hat{j}+18 \hat{k})\) (c) \(\frac{1}{4}(8 \hat{i}+14 \hat{j}+9 \hat{k})\) (d) \(\frac{1}{3}(6 \hat{i}+11 \hat{j}+15 \hat{k})\)

Step-by-Step Solution

Answer

Option (d) \( \frac{1}{3}(6 \hat{i} + 11 \hat{j} + 15 \hat{k}) \).

Step 1: Calculate the vector BC

To find the vector \( \overrightarrow{BC} \), subtract the position vector of \( B \) from \( C \).\[ \overrightarrow{BC} = (2 \hat{i} + 5 \hat{j} + 7 \hat{k}) - (2 \hat{i} + 3 \hat{j} + 4 \hat{k}) = 0 \hat{i} + 2 \hat{j} + 3 \hat{k} \]

Step 2: Calculate the length of sides AB and AC

First, calculate \( \overrightarrow{AB} \) and \( \overrightarrow{AC} \):\[ \overrightarrow{AB} = (2 \hat{i} + 3 \hat{j} + 4 \hat{k}) - (4 \hat{i} + 7 \hat{j} + 8 \hat{k}) = -2 \hat{i} - 4 \hat{j} - 4 \hat{k} \] \[ \overrightarrow{AC} = (2 \hat{i} + 5 \hat{j} + 7 \hat{k}) - (4 \hat{i} + 7 \hat{j} + 8 \hat{k}) = -2 \hat{i} - 2 \hat{j} - 1 \hat{k} \] Find the magnitudes: \[ |\overrightarrow{AB}| = \sqrt{(-2)^2 + (-4)^2 + (-4)^2} = \sqrt{4 + 16 + 16} = \sqrt{36} = 6 \] \[ |\overrightarrow{AC}| = \sqrt{(-2)^2 + (-2)^2 + (-1)^2} = \sqrt{4 + 4 + 1} = \sqrt{9} = 3 \]

Step 3: Apply the section formula

The internal bisector divides \( BC \) in the ratio \( |\overrightarrow{AC}| : |\overrightarrow{AB}| = 3:6 = 1:2 \). Use the section formula to find the position vector \( P \) that divides \( BC \) in the ratio \( 1:2 \):\[ \mathbf{r}_P = \frac{1 \cdot \mathbf{r}_C + 2 \cdot \mathbf{r}_B}{1 + 2} \]Substitute the vectors for \( B \) and \( C \):\[ \mathbf{r}_P = \frac{1 \cdot (2 \hat{i} + 5 \hat{j} + 7 \hat{k}) + 2 \cdot (2 \hat{i} + 3 \hat{j} + 4 \hat{k})}{3} \] Simplify:\[ = \frac{(2 \hat{i} + 5 \hat{j} + 7 \hat{k}) + (4 \hat{i} + 6 \hat{j} + 8 \hat{k})}{3} \]\[ = \frac{6 \hat{i} + 11 \hat{j} + 15 \hat{k}}{3} \ = 2 \hat{i} + \frac{11}{3} \hat{j} + 5 \hat{k} \]

Step 4: Choose the correct option

Find which option matches the calculated position vector. The expression \( \frac{1}{3}(6 \hat{i} + 11 \hat{j} + 15 \hat{k}) \) matches option (d).

Key Concepts

Position Vector

In vector mathematics, a position vector is crucial for describing the location of a point in space with respect to a fixed origin. It is essentially a vector that starts from the origin and ends at a given point. Typically represented in terms of unit vector components—such as \( \hat{i}, \hat{j}, \hat{k} \) in three-dimensional space—a position vector conveys both direction and magnitude.
For instance, in the problem involving triangle \( \triangle ABC \), the position vectors of vertices \( A, B, \) and \( C \) are given as \( 4 \hat{i} + 7 \hat{j} + 8 \hat{k}, 2 \hat{i} + 3 \hat{j} + 4 \hat{k}, \) and \( 2 \hat{i} + 5 \hat{j} + 7 \hat{k} \) respectively. These vectors describe each vertex's position in a spatial coordinate system, providing the foundational data to solve various geometric problems involving the triangle.

Section Formula

The section formula is an essential tool in vector mathematics, particularly useful when determining a point that divides a line segment internally in a given ratio. This formula applies weighted averages of vectors at the segment's end points, defined by the ratio.
In the given exercise, the bisector of angle \( \angle A \) divides the line segment \( BC \) in a specific ratio, which is determined by the magnitudes of sides \( AB \) and \( AC \). Since these magnitudes are calculated as 6 and 3 respectively, the ratio becomes 1:2. Using the section formula, the position vector of this specific point \( P \) on \( BC \) is calculated as:
\[ \mathbf{r}_P = \frac{1 \cdot \mathbf{r}_C + 2 \cdot \mathbf{r}_B}{3} \]
By plugging in the position vectors for \( B \) and \( C \), we find the exact coordinates of the point \( P \), which provides a clear understanding of how the segment is internally divided.

Magnitude of a Vector

The magnitude of a vector, often thought of as the vector's "length," is a critical concept in understanding spatial relationships and distances. It's determined by taking the square root of the sum of the squares of its components. In simpler terms, it represents how far from the origin the point is in a straight line.
In the context of triangle \( \triangle ABC \), the magnitudes are crucial for calculating how the bisector divides \( BC \).

• For \( \overrightarrow{AB} = -2 \hat{i} - 4 \hat{j} - 4 \hat{k} \), the magnitude is\( \sqrt{(-2)^2 + (-4)^2 + (-4)^2} = 6 \).
• For \( \overrightarrow{AC} = -2 \hat{i} - 2 \hat{j} - 1 \hat{k} \), the magnitude is\( \sqrt{(-2)^2 + (-2)^2 + (-1)^2} = 3 \).

These magnitudes help in forming the ratio needed for the section formula, clearly illustrating the role magnitudes play in vector mathematics.

Triangle Geometry

Exploring triangle geometry with vectors opens the door to understanding complex relationships within a triangle using coordinate systems. Using vectors to describe geometry makes problems easier to visualize and solve.
With triangle \( \triangle ABC \), position vectors and their calculations extend to various geometric roles, such as sides and angles. Each side of the triangle can be represented as a vector, and calculations involving these vectors offer insight into internal angles, segment divisions, and bisectors.
The process starts with identifying the position vectors of vertices \( A, B, \) and \( C \). Next, calculating vectors corresponding to triangle sides (like \( \overrightarrow{AB}, \overrightarrow{AC}, \overrightarrow{BC} \)) and their magnitudes helps discover fundamental geometry, such as distances or midpoints.
Furthermore, bisectors and the section formula allow finding specific internal points like the one found where the bisector of \( \angle A \) meets \( BC \). Understanding such geometric principles through vectors highlights a powerful method of navigating mathematical solutions.