Name: Angle Between Two Vectors: Examples
Uploaded: 2019-07-19
Duration: 10 min 7 s
Description: This video explains the proof of the geometric definition of the dot product and also goes over 4 examples illustrating how to find the angle between two vectors.

Question

Prove using vectors: The quadrilateral obtained by joining mid-points of adjacent sides of a rectangle is a rhombus.

Accepted Answer

$A B C D$ is a rectangle.

Let $P, Q, R$ and $S$ be the mid-points of the sides $A B, B C, C D$ and $D A$ , respectively.

$\vec{P B} = \frac{\vec{A B}}{2} = \vec{A P}, \vec{B Q} = \frac{\vec{B C}}{2} = \vec{Q C}, \vec{C R} = \frac{\vec{C D}}{2} = \vec{R D}, \vec{D S} = \frac{\vec{D A}}{2} = \vec{S A}$

Also $A B = C D, B C = A D$ ( $A B C D$ is rectangle opposite sides are equal).

Therefore, $A P = P B = D R = R C$ and $B Q = Q C = A S = S D$ ……(i)

Now, $\vec{P Q} = \vec{P B} + \vec{B Q} = \frac{1}{2} (\vec{A B} + \vec{B C}) = \frac{1}{2} \vec{A C} \dots \dots . (i)$

$\vec{S R} = \vec{S D} + \vec{D R} = \frac{1}{2} (\vec{A D} + \vec{D C}) = \frac{1}{2} \vec{A C} \dots \dots \dots (i i)$

From (i) and (ii), we have

$\vec{P Q} = \vec{S R}$ i.e., sides $P Q$ and $S R$ are equal and parallel.

$P Q R S$ is a parallelogram.

$(\vec{P Q})^{2} = \vec{P Q} \cdot \vec{P Q} = (\vec{P B} + \vec{B Q}) \cdot (\vec{P B} + \vec{B Q}) = (\vec{P B}) \cdot (\vec{P B}) + (\vec{P B}) \cdot (\vec{B Q}) + (\vec{P B}) \cdot (\vec{B Q}) + (\vec{B Q}) \cdot (\vec{B Q}) = | \vec{P B} |^{2} + | \vec{B Q} |^{2} \dots \dots . . (i i i) \{(\vec{P B}) \cdot (\vec{B Q}) = 0, (\vec{P B}) ⊥ (\vec{B Q})\}$

Similarly, $(\vec{P S})^{2} = \vec{P S} \cdot \vec{P S} = (\vec{P A} + \vec{A S}) \cdot (\vec{P A} + \vec{A S}) = | \vec{P A} |^{2} + | \vec{A S} |^{2} = | \vec{P B} |^{2} + | \vec{B Q} |^{2} . . . . . . (i v)$

From (iii) and (iv) we get, $(\vec{P Q})^{2} = (\vec{P S})^{2} i.e., P Q = P S$ .

The adjacent sides of $P Q R S$ are equal. Therefore, $P Q R S$ is a rhombus.

Prove using vectors: The quadrilateral obtained by joining mid-points of adjacent sides of a rectangle is a rhombus.

Important Questions on Scalar or Dot Product

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