Question

The vectors \[\bar A\vec B = 3\hat i + 5j + 4\hat k\] and \[\bar A\vec C = 5\hat i - 5j + 2\hat k\] are the sides of a triangle ABC. The length of the median through \[A\] is
A. \[3\]
B. \[ - 4\]
C. \[1\]
D. \[5\]

Answer 1

Hint: In the given question, we need to find the value of the length of the median through \[A\]. For this, we will find the position vector of \[AD\] and its modulus to get the desired result. Also, we will draw the diagram for this.

Formula used: The following formula used for solving the given question.
The modulus of a vector \[\vec u = a\hat i + b\hat j + c\hat k\] is given by
\[\left| {\vec u} \right| = \sqrt {{a^2} + {b^2} + {c^2}} \]

Complete step by step solution: Consider the following figure.

We know that the vectors \[\bar A\vec B = 3\hat i + 5j + 4\hat k\] and \[\bar A\vec C = 5\hat i - 5j + 2\hat k\] are the sides of a triangle ABC.
Here, we will find the position vector of \[AD\]
Thus, the position vector is given by
\[AD = \dfrac{{\left( {3 + 5} \right)\hat i + \left( {5 - 5} \right)\hat j + \left( {4 + 2} \right)\hat k}}{2}\]
By simplifying, we get
\[AD = \dfrac{{\left( 8 \right)\hat i + \left( 0 \right)\hat j + \left( 6 \right)\hat k}}{2}\]
By simplifying, we get
\[AD = \left( 4 \right)\hat i + \left( 3 \right)\hat k\]
Now, the length of a medium is given by
\[\left| {AD} \right| = \sqrt {{{\left( 4 \right)}^2} + {{\left( 3 \right)}^2}} \]
\[\left| {AD} \right| = \sqrt {16 + 9} \]
\[\left| {AD} \right| = \sqrt {25} \]
By taking square root, we get
\[\left| {AD} \right| = 5\]
Hence, the length of the median through \[A\] is \[5\] if the vectors \[\bar A\vec B = 3\hat i + 5j + 4\hat k\] and \[\bar A\vec C = 5\hat i - 5j + 2\hat k\] are the sides of a triangle ABC.

Thus, Option (D) is correct.

Additional Information:The definition of a vector is an entity with both magnitude and direction. The movement of an object between two points is described by a vector. The directed line segment can be used to geometrically represent vector mathematics. The magnitude of a vector is the length of the directed line segment, and the vector's direction is indicated by the angle at which it is inclined. A vector's "Tail" (the point where it begins) and "Head" (the point where it ends and has an arrow) are its respective names.

Note: Many students make mistake in finding the position vector of \[AD\]. This is the only way, through which we can solve the example in simplest way. Also, it is necessary to draw the figure to analyse the question to get the desired result.