The area of the square, one of whose diagonals is 3î + 4ĵ is
12.5 square unit
Understanding the Problem: Area of a Square from Diagonal Vector
The question asks us to find the area of a square when we are given one of its diagonals as a vector. The diagonal is represented by the vector $\vec{d} = 3\hat{i} + 4\hat{j}$. To find the area of the square, we need to determine the length of this diagonal.
Calculating the Length of the Diagonal
A vector in the form $a\hat{i} + b\hat{j}$ represents a displacement from the origin to the point (a, b). The magnitude (or length) of this vector is calculated using the Pythagorean theorem.
Given the diagonal vector $\vec{d} = 3\hat{i} + 4\hat{j}$, the components are $a=3$ and $b=4$.
The magnitude of the diagonal vector, which is the length of the diagonal (let's call it $D$), is given by:
$$D = \sqrt{a^2 + b^2}$$
Substituting the values:
$$D = \sqrt{(3)^2 + (4)^2}$$
$$D = \sqrt{9 + 16}$$
$$D = \sqrt{25}$$
$$D = 5 \text{ units}$$
So, the length of the diagonal of the square is 5 units.
Finding the Area of the Square
There are two common formulas for the area of a square:
- Area = side length $\times$ side length (side²)
- Area = $\frac{1}{2} \times$ diagonal length $\times$ diagonal length ($\frac{1}{2}D^2$)
Since we have calculated the length of the diagonal, the second formula is more direct.
Using the diagonal length $D = 5$ units:
$$ \text{Area} = \frac{1}{2} \times D^2 $$
$$ \text{Area} = \frac{1}{2} \times (5)^2 $$
$$ \text{Area} = \frac{1}{2} \times 25 $$
$$ \text{Area} = \frac{25}{2} $$
$$ \text{Area} = 12.5 \text{ square units} $$
Thus, the area of the square with a diagonal vector $3\hat{i} + 4\hat{j}$ is 12.5 square units.
Summary of Steps to find Square Area from Diagonal Vector
- Identify the components of the diagonal vector, say $a\hat{i} + b\hat{j}$.
- Calculate the magnitude of the vector using $\sqrt{a^2 + b^2}$ to find the diagonal length, $D$.
- Use the formula Area = $\frac{1}{2}D^2$ to find the area of the square.
| Concept | Formula/Method |
|---|---|
| Magnitude of vector $a\hat{i} + b\hat{j}$ | $\sqrt{a^2 + b^2}$ |
| Area of a square given diagonal $D$ | $\frac{1}{2}D^2$ |
Revision Table: Key Formulas for Square Properties
| Property | Formula (Side = $s$, Diagonal = $D$) |
|---|---|
| Side Length | $s$ |
| Diagonal Length | $D = s\sqrt{2}$ |
| Area | $s^2$ or $\frac{1}{2}D^2$ |
| Perimeter | $4s$ |
Additional Information: Vectors and Geometric Shapes
Vectors can be used to represent lengths and directions in geometry. The magnitude of a vector gives its length.
- A vector $a\hat{i} + b\hat{j}$ can represent a side or a diagonal of a shape.
- The magnitude $\sqrt{a^2 + b^2}$ is the length of the line segment represented by the vector.
- In this problem, the vector represents the diagonal of the square. The length of this vector is the length of the diagonal.
- Knowing the diagonal length is sufficient to calculate the area of a square.
Understanding vector magnitudes is crucial for solving geometric problems involving vectors.
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