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Chapter 1: Problem 25

If a unit vector is represented by \(0.5 \hat{i}+0.8 \hat{j}+c \hat{k}\) the value of \(c\) is (A) 1 (B) \(\sqrt{0.11}\) (C) \(\sqrt{0.01}\) (D) \(0.39\)

Short Answer

Expert verified
The value of 'c' in the unit vector \(0.5 \hat{i} + 0.8 \hat{j} + c \hat{k}\) is \(c = \sqrt{0.11}\) (Option B).

Step by step solution

01

Calculate the Magnitude of the Vector

To find the magnitude of the given vector, we will use the formula \(\| \vec{V} \|= \sqrt {(0.5)^2 + (0.8)^2 + c^2}\), where \(\vec{V}\) is the given vector and c is the unknown component in \(\hat{k}\) direction.
02

Set the magnitude equal to 1

Since the vector is a unit vector, its magnitude will be equal to 1. So, \(\| \vec{V} \| = 1\) We can now set the expression from Step 1 equal to 1 and solve for c. \(\sqrt {(0.5)^2 + (0.8)^2 + c^2} = 1\)
03

Solve for c

To solve for c, we first square both sides of the equation: \((0.5)^2 + (0.8)^2 + c^2 = 1^2\) Now, subtract \((0.5)^2 +(0.8)^2\) from both sides to isolate the \(c^2\) term: \(c^2 = 1 - 0.25 - 0.64\) Calculate the value of c^2: \(c^2 = 0.11\) Finally, take the square root of both sides to find the value of c: \(c = \sqrt{0.11}\)
04

Choose the correct answer

The value of 'c' is found to be \(\sqrt{0.11}\), which corresponds to option (B). Therefore, the correct answer is: \(c = \sqrt{0.11}\) (Option B)

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