Name: Parallelogram Law for Addition of Two Vectors
Uploaded: 2019-07-19
Duration: 5 min 1 s
Description: Now, we know one of the methods to add the two vectors. It is time to learn another law to add the two vectors. Explore this video to learn about it.

Question

Prove by vector method that: the diagonals of a parallelogram bisect each other.

Accepted Answer

Let $A$ be at origin and $\vec{A B} = \vec{a} and \vec{A D} = \vec{b}$

Again, let $A C and B D$ intersect each other at $H$ .

We have to prove that $H$ is middle point of $A C and B D$

Let $\vec{A H} = x \vec{A C} and \vec{H B} = y \vec{D B} \to (1)$

Now, $\vec{A C} = \vec{A B} + \vec{B C} = \vec{A B} + \vec{A D} = \vec{a} + \vec{b}$ [Using triangle law of addition]

Also, $\vec{D B} = \vec{D A} + \vec{A B} = \vec{A B} - \vec{A D} = \vec{a} - \vec{b}$

Therefore, $\vec{A H} = x (\vec{a} + \vec{b}) and \vec{H B} = y (\vec{a} - \vec{b})$ [Using $(1)$ ]

Now, $\vec{A B} = \vec{A H} + \vec{H B}$

$\Rightarrow \vec{a} = x (\vec{a} + \vec{b}) + y (\vec{a} - \vec{b})$

$\Rightarrow \vec{a} = x \vec{a} + x \vec{b} + y \vec{a} - y \vec{b}$

$\Rightarrow \vec{a} = (x + y) \vec{a} + (x - y) \vec{b}$

Equating the coefficients of $\vec{a} and \vec{b}$ , then we obtain: $x + y = 1 and x - y = 0$

Therefore, $2 x = 1 \Rightarrow x = \frac{1}{2}$ and $y = \frac{1}{2}$

So, from $(1)$ , we have; $\vec{A H} = \frac{1}{2} \vec{A C} and \vec{H B} = \frac{1}{2} \vec{D B}$

Hence, $H$ is the middle point of $A C and B D$ or diagonals of parallelogram bisect each other.

Prove by vector method that: the diagonals of a parallelogram bisect each other.

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