The sum and diffrence of two perpendicular vector of equal length are

Options

A. Perpendicular to each other and of equal length.
B. Perpendicular to each other and of different length
C. Of equal length and have an obtuse angle between them
D. Of equal length and have an acute angle between them

Explanation

To solve the problem of finding the sum and difference of two perpendicular vectors of equal length, we can follow these steps:

### Step 1: Define the Vectors
Let the two vectors be \( \vec{A} \) and \( \vec{B} \), both having equal magnitudes. We can denote their magnitudes as \( A = B = a \).

### Step 2: Express the Vectors
Since the vectors are perpendicular, we can represent them in a Cartesian coordinate system:
- Let \( \vec{A} = a \hat{i} \) (along the x-axis)
- Let \( \vec{B} = a \hat{j} \) (along the y-axis)

### Step 3: Calculate the Sum of the Vectors
The sum of the two vectors is given by:
\[ \vec{C} = \vec{A} + \vec{B} = a \hat{i} + a \hat{j} \]
To find the magnitude of \( \vec{C} \):
\[ |\vec{C}| = \sqrt{(a)^2 + (a)^2} = \sqrt{2a^2} = a\sqrt{2} \]

### Step 4: Calculate the Difference of the Vectors
The difference of the two vectors is given by:
\[ \vec{D} = \vec{A} - \vec{B} = a \hat{i} - a \hat{j} \]
To find the magnitude of \( \vec{D} \):
\[ |\vec{D}| = \sqrt{(a)^2 + (-a)^2} = \sqrt{a^2 + a^2} = \sqrt{2a^2} = a\sqrt{2} \]

### Step 5: Analyze the Results
Both the sum \( \vec{C} \) and the difference \( \vec{D} \) have the same magnitude of \( a\sqrt{2} \). Additionally, since \( \vec{C} \) and \( \vec{D} \) are formed from the addition and subtraction of two perpendicular vectors, they are also perpendicular to each other.

### Conclusion
The sum and difference of two perpendicular vectors of equal length are:
- Magnitudes: \( |\vec{C}| = a\sqrt{2} \) and \( |\vec{D}| = a\sqrt{2} \)
- They are perpendicular to each other.

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