Problem 27 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

The vectors \(\overrightarrow{A B}=3 \hat{i}+4 \hat{k}\) \& \(\overrightarrow{A C}=5 \hat{i}-2 \hat{j}+4 \hat{k}\) are the sides of a triangle \(\mathrm{ABC}\). The length of the median through A is (a) \(\sqrt{288}\) (b) \(\sqrt{18}\) (c) \(\sqrt{72}\) (d) \(\sqrt{33}\)

Step-by-Step Solution

Answer

The length of the median through A is \( \sqrt{33} \). Option (d).

Step 1: Find Midpoint D of BC

To find the median through point A, we first need to find the midpoint D of side BC of triangle ABC. Given \( \overrightarrow{AB} = 3 \hat{i} + 4 \hat{k} \) and \( \overrightarrow{AC} = 5 \hat{i} - 2 \hat{j} + 4 \hat{k} \), the vector \( \overrightarrow{BC} \) can be computed as \( \overrightarrow{AC} - \overrightarrow{AB} = (5\hat{i} - 2\hat{j} + 4\hat{k}) - (3\hat{i} + 4\hat{k}) = 2\hat{i} - 2\hat{j} \).The midpoint D of vector BC is half the vector, hence \( \overrightarrow{BD} = \frac{1}{2}(2\hat{i} - 2\hat{j}) = \hat{i} - \hat{j} \). Factors through C to find D are negated, hence D is \( \overrightarrow{A} + (3\hat{i} + 4\hat{k}) + \overrightarrow{BD} = 4\hat{i} - \hat{j} + 4\hat{k} \)

Step 2: Find Vector AD

We now compute \( \overrightarrow{AD} \) using points A and D. A *coincides* with the origin \( A = \overrightarrow{0} \). From step 1, we have found D = \( 4\hat{i} - \hat{j} + 4\hat{k} \).Hence, the vector \( \overrightarrow{AD} = 4\hat{i} - \hat{j} + 4\hat{k} \) assuming origin \( O = 0 \hat{i} + 0 \hat{j} + 0 \hat{k} \) for \( \overrightarrow{A} \).

Step 3: Calculate the Magnitude of AD

To find the length of the vector \( \overrightarrow{AD} \), compute its magnitude using the formula:\[|\overrightarrow{AD}| = \sqrt{(4)^2 + (-1)^2 + (4)^2} = \sqrt{16 + 1 + 16} = \sqrt{33}\]

Step 4: Select the Correct Answer

The calculated length of the median through A (|\overrightarrow{AD}|) is \( \sqrt{33} \). Therefore, the correct answer is option (d) \( \sqrt{33} \).

Key Concepts

Triangle Median

In the context of a triangle, a median is a crucial concept. It is a line segment that connects a vertex of the triangle to the midpoint of the opposite side. This segment divides the triangle into two smaller triangles of equal area. The median can be visualized as a balancing line, giving a sense of symmetry within the triangle.

If we consider a triangle ABC, the task is to find the median passing through a specific vertex, such as point A. To do this, we first need to identify the midpoint of the opposite side, BC in this case. The midpoint provides the destination point for our median.

• Step 1: Calculate the vector for the opposite side, BC.
• Step 2: Determine its midpoint, which is the average position of its endpoints.

These steps help us to find the median’s path from A, ultimately allowing us to compute or evaluate other properties like its length.

Vector Magnitude

The magnitude of a vector essentially measures its length. It's calculated using the vector components provided by the vector equation. Think of it as determining how "long" or "far" the vector stretches in space, similar to finding the distance between two points.

For a vector given as \( \overrightarrow{V} = a \hat{i} + b \hat{j} + c \hat{k} \), its magnitude \(|\overrightarrow{V}|\) is defined as:\[|\overrightarrow{V}| = \sqrt{a^2 + b^2 + c^2}\]
This formula stems from the Pythagorean theorem in three-dimensional space, allowing us to compute the absolute size of the vector irrespective of its direction. The components \( a, b, \) and \( c \) square and sum to reflect the vector's displacement along each axis.

In our exercise, the vector \( \overrightarrow{AD} = 4\hat{i} - \hat{j} + 4\hat{k} \) has a calculated magnitude of \( \sqrt{33} \), demonstrating how these steps work in practice.

Coordinate Geometry

Coordinate geometry, sometimes termed analytic geometry, is the study of geometry using a coordinate system. This framework uses algebra to describe geometric concepts, linking algebraic equations to geometrical figures like lines and points.

In problems involving vectors and triangles, coordinate geometry helps in performing calculations by allowing us to handle vectors in terms of their components. For example, vectors like \( \overrightarrow{AB} \) and \( \overrightarrow{AC} \) are expressed in components along the x, y, and z axes. This enables straightforward calculations for operations such as finding midpoints or vector differences.

• Representation of vectors in i, j, and k simplifies dealing with spatial positions.
• Midpoint calculations benefit from averaging the respective coordinates.
• Coordinate geometry succinctly ties vector operations and geometric interpretations together.

Using coordinate geometry, we not only explore shapes but are also able to perform precise numerical operations, turning a geometric problem into a solvable algebraic task.