Problem 106 - A Complete Resource Book in Mathematics for JEE Main

Question

If the vectors \(\overrightarrow{A B}=3 \hat{i}+4 \hat{k}\) and \(\overrightarrow{A C}=5 \hat{i}+2 \hat{j}+4 \hat{k}\) represent the sides of a triangle \(A B C\), then the length of the median through \(A\) is [2013] (A) \(\sqrt{72}\) (B) \(\sqrt{33}\) (C) \(\sqrt{45}\) (D) \(\sqrt{18}\)

Step-by-Step Solution

Answer

Option (B) \(\sqrt{33}\).

Step 1: Calculate the Midpoint of BC

To find the midpoint of the line segment BC, we use the formula for the midpoint: \[ M = \frac{\overrightarrow{AC} + \overrightarrow{AB}}{2} \]. Hence, \[ M = \frac{(5\hat{i} + 2\hat{j} + 4\hat{k}) + (3\hat{i} + 4\hat{k})}{2} = \frac{(8\hat{i} + 2\hat{j} + 8\hat{k})}{2} = 4\hat{i} + \hat{j} + 4\hat{k} \].

Step 2: Find the Vector AM

The vector \(\overrightarrow{AM}\) is given by subtracting \(\overrightarrow{A}\) from the midpoint \(\overrightarrow{M}\). Assuming that \(\overrightarrow{A}\) is the origin, \(\overrightarrow{AM} = \overrightarrow{M} = 4\hat{i} + \hat{j} + 4\hat{k} \).

Step 3: Calculate the Length of the Median AM

The length of \(\overrightarrow{AM}\) is the magnitude of the vector. Use the formula for the magnitude: \[ |\overrightarrow{AM}| = \sqrt{(4)^2 + (1)^2 + (4)^2} = \sqrt{16 + 1 + 16} = \sqrt{33} \].

Step 4: Compare with Given Options

Among the given options, option (B) \(\sqrt{33}\) matches the calculated length of the median \(\overrightarrow{AM}\).

Key Concepts

Midpoint Formula

The midpoint formula is a crucial concept in vector geometry. It helps in finding the point that is exactly halfway between two defined points in space. This point is known as the "midpoint." The formula for calculating the midpoint of a vector between points is

• \( M = \frac{\overrightarrow{P} + \overrightarrow{Q}}{2} \)

where \( M \) is the midpoint, and \( \overrightarrow{P} \) and \( \overrightarrow{Q} \) are the position vectors of points \( A \) and \( B \), respectively.
In the exercise provided, you find the midpoint \( M \) of the line segment \( BC \) by adding the vectors \( \overrightarrow{AB} \) and \( \overrightarrow{AC} \), before dividing the result by two.
An easy way to think of the midpoint is like finding the average of two points. This technique doesn’t just apply to the two-dimensional plane; it extends to three-dimensions and beyond. This utility is especially helpful in problems involving triangles in vector geometry, allowing us to find an exact center for medians.

Vector Magnitude

Vector magnitude, often referred to as the length or norm of a vector, measures the size of the vector. It can be calculated using the Euclidean distance formula.
For a vector \( \overrightarrow{v} = x \hat{i} + y \hat{j} + z \hat{k} \), the magnitude is found using the formula:

• \( |\overrightarrow{v}| = \sqrt{x^2 + y^2 + z^2} \)

This equation uses the components of the vector, each squared, summed, and square rooted, similar to finding the hypotenuse in the Pythagorean theorem, but in three dimensions rather than just two.
In the exercise, calculating the magnitude of \( \overrightarrow{AM} \) involves finding the magnitude of the given vector \( 4\hat{i} + \hat{j} + 4\hat{k} \), resulting in a length of \( \sqrt{33} \). Understanding how to determine vector magnitude is vital in multiple fields, from physics to computer graphics, where knowing the "size" or "length" of a movement vector is essential.

Triangle Median

In geometry, a median of a triangle is a line segment joining a vertex to the midpoint of the opposite side. Medians are significant because they coincide at a single point known as the centroid, which is the center of mass of the triangle. This balanced center divides each median into two segments, one of which is double the length of the other.
To find the median through a given point, the midpoint of the opposite side is first determined, which is why the midpoint formula is so essential in this calculation.
The length of the median from a vertex \( A \) to the midpoint \( M \) of side \( BC \) is computed by first finding \( M \), then using \( \overrightarrow{AM} \) for the magnitude calculation. The exercise above reflects this process as the final length of the median \( \overrightarrow{AM} \) comes out to be \( \sqrt{33} \), emphasizing the harmony between each calculation stage, from using the midpoint formula to applying vector magnitude.