Question

The area of rectangle with sides as \(\vec A=3\hat i+4\hat j\) and \(\vec B=\hat i+3\hat j\) is

• A. \(5\sqrt{10}\) units
• B. \(10\) units
• C. \(2\sqrt{10}\) units
• D. \(10\sqrt5\) units

Hint:

Magnitude of \(a\hat i+b\hat j\) is \(\sqrt{a^2+b^2}\). For rectangle area, multiply the magnitudes of adjacent sides.

Answer 1

The Correct Option is A

We are given two side vectors of a rectangle: \[ \vec A=3\hat i+4\hat j \] and \[ \vec B=\hat i+3\hat j. \] The area of a rectangle is: \[ \text{Area}=\text{length}\times \text{breadth}. \] Here, length and breadth are magnitudes of the vectors. First find magnitude of \(\vec A\): \[ |\vec A|=\sqrt{3^2+4^2}. \] \[ |\vec A|=\sqrt{9+16}. \] \[ |\vec A|=\sqrt{25}=5. \] Now find magnitude of \(\vec B\): \[ |\vec B|=\sqrt{1^2+3^2}. \] \[ |\vec B|=\sqrt{1+9}. \] \[ |\vec B|=\sqrt{10}. \] Therefore, area is: \[ \text{Area}=|\vec A||\vec B|. \] \[ =5\cdot \sqrt{10}. \] \[ =5\sqrt{10}. \] Hence, the area of the rectangle is: \[ 5\sqrt{10}\text{ units}. \]