If a unit vector is represented by `0.5hati + 0.8hatj + chatk` the value of c is

Options

A. 1
B. `sqrt(0.11)`
C. `sqrt(0.01)`
D. 0.39

Explanation

To find the value of \( c \) in the unit vector represented by \( 0.5 \hat{i} + 0.8 \hat{j} + c \hat{k} \), we need to follow these steps:

### Step 1: Understand the Definition of a Unit Vector
A unit vector has a magnitude of 1. Therefore, we need to calculate the magnitude of the given vector and set it equal to 1.

### Step 2: Write the Expression for the Magnitude
The magnitude of a vector \( \mathbf{A} = A_x \hat{i} + A_y \hat{j} + A_z \hat{k} \) is given by:
\[ |\mathbf{A}| = \sqrt{A_x^2 + A_y^2 + A_z^2} \]
In our case:
\[ |\mathbf{A}| = \sqrt{(0.5)^2 + (0.8)^2 + c^2} \]

### Step 3: Substitute the Values
Substituting the values into the magnitude formula, we get:
\[ |\mathbf{A}| = \sqrt{(0.5)^2 + (0.8)^2 + c^2} = \sqrt{0.25 + 0.64 + c^2} \]

### Step 4: Set the Magnitude Equal to 1
Since \( \mathbf{A} \) is a unit vector, we set the magnitude equal to 1:
\[ \sqrt{0.25 + 0.64 + c^2} = 1 \]

### Step 5: Square Both Sides
To eliminate the square root, we square both sides:
\[ 0.25 + 0.64 + c^2 = 1^2 \]
This simplifies to:
\[ 0.25 + 0.64 + c^2 = 1 \]

### Step 6: Combine Like Terms
Now, combine the constants:
\[ 0.89 + c^2 = 1 \]

### Step 7: Isolate \( c^2 \)
Subtract \( 0.89 \) from both sides:
\[ c^2 = 1 - 0.89 \]
\[ c^2 = 0.11 \]

### Step 8: Solve for \( c \)
Taking the square root of both sides gives us:
\[ c = \sqrt{0.11} \]
Thus, \( c \) can be either positive or negative:
\[ c = \pm \sqrt{0.11} \]

### Final Answer
The value of \( c \) is \( \sqrt{0.11} \) or \( -\sqrt{0.11} \).

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