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If a unit vector is represented by 0.5 h...

If a unit vector is represented by `0.5 hat(i) + 0.8 hat(j) + c hat(k)` , then the value of `c` is

A

`1`

B

`sqrt(0.11)`

C

`sqrt(0.01)`

D

`sqrt(0.39)`

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The correct Answer is:
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 ensure that the magnitude of this vector is equal to 1, as it is a unit vector. ### Step-by-Step Solution: 1. **Understand the Magnitude of a Vector**: The magnitude of a vector \( \vec{A} = a \hat{i} + b \hat{j} + c \hat{k} \) is given by the formula: \[ |\vec{A}| = \sqrt{a^2 + b^2 + c^2} \] 2. **Identify Components**: In our case, the components are: - \( a = 0.5 \) - \( b = 0.8 \) - \( c = c \) 3. **Set Up the Equation for Magnitude**: Since it is a unit vector, we have: \[ \sqrt{(0.5)^2 + (0.8)^2 + c^2} = 1 \] 4. **Square Both Sides**: To eliminate the square root, square both sides of the equation: \[ (0.5)^2 + (0.8)^2 + c^2 = 1^2 \] This simplifies to: \[ 0.25 + 0.64 + c^2 = 1 \] 5. **Combine the Known Values**: Now, combine \( 0.25 \) and \( 0.64 \): \[ 0.89 + c^2 = 1 \] 6. **Isolate \( c^2 \)**: Subtract \( 0.89 \) from both sides: \[ c^2 = 1 - 0.89 \] \[ c^2 = 0.11 \] 7. **Take the Square Root**: To find \( c \), take the square root of both sides: \[ c = \sqrt{0.11} \] 8. **Final Value of \( c \)**: Thus, the value of \( c \) is: \[ c \approx 0.3317 \] ### Summary: The value of \( c \) in the unit vector \( 0.5 \hat{i} + 0.8 \hat{j} + c \hat{k} \) is \( \sqrt{0.11} \approx 0.3317 \).

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 ensure that the magnitude of this vector is equal to 1, as it is a unit vector. ### Step-by-Step Solution: 1. **Understand the Magnitude of a Vector**: The magnitude of a vector \( \vec{A} = a \hat{i} + b \hat{j} + c \hat{k} \) is given by the formula: \[ |\vec{A}| = \sqrt{a^2 + b^2 + c^2} ...
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Knowledge Check

  • If a unit vector is represented by 0.5 hat(i)-0.8 hat(j)+chat(k) , then the value of 'c' is :-

    A
    `1`
    B
    `sqrt(0.11)`
    C
    `sqrt(0.01)`
    D
    `sqrt(0.39)`
  • A unit vector is represented as (0.8hat(i)+bhat(j)+04hat(k)) . Hence, the value of b must be

    A
    0.4
    B
    `sqrt(0.6)`
    C
    0.2
    D
    `sqrt(0.2)`
  • A unit vector is represented as (0.8hat(i)+bhat(j)+0.4hat(k)) . Hence the value of 'b' must be

    A
    0.4
    B
    `sqrt0.6`
    C
    0.2
    D
    `sqrt0.2`
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