Question

ABCD is a parallelogram. Show that 2DC-DB= AC.

Answer 1

Hint: To solve this question given to us we have to use the triangle law of vector addition and some properties of parallelogram like we know that the opposite sides of parallelogram are equal in order to proof the above given equation in the parallelogram.

Complete answer:

In the parallelogram given above according to the property of parallelogram that is the opposite sides of parallelogram are equal. we can say that,
AB = DC (opposite sides of parallelogram)
AD = BC (opposite sides of parallelogram)
We have to proof that 2DC – DB = AC
Now, we will use the triangle law of vector addition which says that when two vectors are represented as the two sides of the triangle with the order of magnitude and direction, then the third side of the triangle represents the magnitude and the direction of the resultant vector.
So, if we use the triangle law of vector addition here, we can clearly say that,
DA + AB = DB which will form equation 1
And also,
-DA + DC = AC (here we have used -DA instead of DA because of the opposite direction of vector that is the negative direction of the vector)
So, we can say that, DC = AC + DA which will form equation 2
Now, we are given that,
 LHS in the question is 2DC – DB
=2(AC + DA) – (DA + AB) from equation 1 and equation 2
=2AC + 2DA – DA – AB
=2AC + DA – AB
=2AC + DA – CD (as we know opposite sides of parallelogram are equal so AB = CD)
=2AC + (-AC) from equation 2 which says that DC = AC + DA
= AC
LHS = RHS
Hence, we can say that in the given parallelogram ABCD, 2DC – DB = AC.

Note:
In the questions like the one given here always take care of the directions of the vectors given in the questions as you can usually make mistakes if you don’t keep the account of the direction and the answer will not be correct that you will find that way.