If a unit vector is represented by `0.4 hati+0.7 hatj + chatk` then the value of c is

Options

A. `sqrt(0.01)`
B. `sqrt(0.11)`
C. 1
D. `sqrt(0.35)`

Answer

To find the value of \( c \) in the unit vector represented by \( 0.4 \hat{i} + 0.7 \hat{j} + c \hat{k} \), we can follow these steps:

### Step 1: Understand the properties of a unit vector
A unit vector has a magnitude of 1. Therefore, we can express the magnitude of the vector \( \vec{n} = 0.4 \hat{i} + 0.7 \hat{j} + c \hat{k} \) as follows:

\[ |\vec{n}| = \sqrt{(0.4)^2 + (0.7)^2 + c^2} \]

### Step 2: Set up the equation for the magnitude
Since \( \vec{n} \) is a unit vector, we set its magnitude equal to 1:

\[ \sqrt{(0.4)^2 + (0.7)^2 + c^2} = 1 \]

### Step 3: Square both sides to eliminate the square root
Squaring both sides gives us:

\[ (0.4)^2 + (0.7)^2 + c^2 = 1^2 \]

### Step 4: Calculate \( (0.4)^2 \) and \( (0.7)^2 \)
Calculating the squares:

\[ (0.4)^2 = 0.16 \]
\[ (0.7)^2 = 0.49 \]

### Step 5: Substitute the values into the equation
Now substitute these values back into the equation:

\[ 0.16 + 0.49 + c^2 = 1 \]

### Step 6: Simplify the equation
Adding \( 0.16 \) and \( 0.49 \):

\[ 0.65 + c^2 = 1 \]

### Step 7: Solve for \( c^2 \)
Subtract \( 0.65 \) from both sides:

\[ c^2 = 1 - 0.65 \]
\[ c^2 = 0.35 \]

### Step 8: Solve for \( c \)
Taking the square root of both sides gives:

\[ c = \sqrt{0.35} \]

### Conclusion
Thus, the value of \( c \) is \( \sqrt{0.35} \).

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