Question

ABCD is a parallelogram. If L and M are the middle points of BC and CD, then what is the value of \[\overrightarrow {AL} + \overrightarrow {AM} \] ?
(a). \[\dfrac{1}{2}\overrightarrow {AC} \]
(b). \[\dfrac{3}{2}\overrightarrow {AC} \]
(c). \[\overrightarrow {AC} \]
(d). \[\dfrac{2}{3}\overrightarrow {AC} \]

Answer 1

Hint: Express the vectors \[\overrightarrow {AL} \] and \[\overrightarrow {AM} \] in terms of the sides of the parallelogram. The vector AC is the sum of the vectors of adjacent sides of the triangle, that is, \[\overrightarrow {AB} + \overrightarrow {BC} \] and \[\overrightarrow {AD} + \overrightarrow {DC} \]. Then, express the sum \[\overrightarrow {AL} + \overrightarrow {AM} \] in terms of \[\overrightarrow {AC} \].

Complete step-by-step answer:
A parallelogram is a special type of quadrilateral whose opposite sides are parallel and equal to each other.
A vector is a quantity that has both magnitude and direction. A vector is free to be dragged parallel to it. Hence, two vectors having equal magnitude and pointing in the same direction are equal.
The sum of the vectors of the adjacent sides of the parallelogram gives the diagonal of the parallelogram.

We have the parallelogram ABCD. L and M are midpoints of the sides BC and CD respectively.
We need to find the sum of the vectors \[\overrightarrow {AL} \] and \[\overrightarrow {AM} \].
The vector AL can be written as the sum of the vectors AB and BL.
\[\overrightarrow {AL} = \overrightarrow {AB} + \overrightarrow {BL} ..........(1)\]
The vector AM can be written as the sum of the vectors AD and DM.
\[\overrightarrow {AM} = \overrightarrow {AD} + \overrightarrow {DM} ..........(2)\]
Adding equation (1) and (2), we have:
\[\overrightarrow {AL} + \overrightarrow {AM} = \overrightarrow {AB} + \overrightarrow {BL} + \overrightarrow {AD} + \overrightarrow {DM} ..........(3)\]
We know that vector AD is equal to vector BC since they are equal in magnitude and direction.
\[\overrightarrow {AD} = \overrightarrow {BC} ............(4)\]
We also know that the vector BL is half of the vector BC and the vector DM is half of the vector DC.
\[\overrightarrow {BL} = \dfrac{1}{2}\overrightarrow {BC} ..........(5)\]
\[\overrightarrow {DM} = \dfrac{1}{2}\overrightarrow {DC} \]
The vector DC is equal to the vector AB since they are equal in magnitude and direction.
\[\overrightarrow {DM} = \dfrac{1}{2}\overrightarrow {AB} ............(6)\]
Using equations (4), (5), and (6) in equation (3), we get:
\[\overrightarrow {AL} + \overrightarrow {AM} = \overrightarrow {AB} + \dfrac{1}{2}\overrightarrow {BC} + \overrightarrow {BC} + \dfrac{1}{2}\overrightarrow {AB} \]

Simplifying, we get:
\[\overrightarrow {AL} + \overrightarrow {AM} = \dfrac{3}{2}\overrightarrow {AB} + \dfrac{3}{2}\overrightarrow {BC} \]
\[\overrightarrow {AL} + \overrightarrow {AM} = \dfrac{3}{2}(\overrightarrow {AB} + \overrightarrow {BC} )...........(7)\]
The vector AC is the sum of the vectors AB and BC, hence we have:
\[\overrightarrow {AC} = \overrightarrow {AB} + \overrightarrow {BC} ...........(8)\]
Using equation (8) in equation (7), we get:
\[\overrightarrow {AL} + \overrightarrow {AM} = \dfrac{3}{2}\overrightarrow {AC} \]
Hence, option (b) is the correct answer.

Note: You need to be careful when doing vector addition, only vectors parallel to each other can be added and subtracted directly. For example, \[\overrightarrow {AC} \] and \[\overrightarrow {AB} \] can not be added directly.