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Chapter 2: Problem 290

Find a unit vector in direction of \(\hat{1}+2 \hat{\jmath}-3 \mathrm{k}\) (A) \((1 / \sqrt{7})(\hat{1}+2 \hat{\jmath}-3 \mathrm{k})\) (B) \(-(1 / 2)(\hat{1}+2 \hat{\jmath}-3 \mathrm{k})\) (C) \((1 / \sqrt{14})(\hat{1}+2 \hat{j}-3 \mathrm{k})\) (D) \((1 / \sqrt{5})(\hat{1}+2 \hat{\jmath}-3 \mathrm{k})\)

Short Answer

Expert verified
(C) \((1 / \sqrt{14})(\hat{1}+2 \hat{j}-3 \mathrm{k})\)

Step by step solution

01

Find the magnitude of the given vector

To find the magnitude of the given vector, we can use the formula: \[|\vec{v}| = \sqrt{\vec{v} \cdot \vec{v}}\] where \(\cdot\) represents the dot product. In this case, we have: \[\vec{v} = \hat{1}+2 \hat{\jmath}-3 \mathrm{k}\] Calculate the dot product of \(\vec{v}\) with itself: \[\vec{v} \cdot \vec{v} = (1)^{2} +(2)^{2} +(-3)^{2} = 1 + 4 + 9 = 14\] \squareCalculate the magnitude of \(\vec{v}\) by taking the square root of the dot product: \[|\vec{v}| = \sqrt{14}\]
02

Divide the given vector by its magnitude

Now we will divide the given vector by its magnitude to find the unit vector: \[\frac{\vec{v}}{|\vec{v}|} = \frac{\hat{1}+2 \hat{\jmath}-3 \mathrm{k}}{\sqrt{14}}\] The unit vector in the direction of \(\vec{v}\) is: \[\frac{1}{\sqrt{14}}(\hat{1}+2 \hat{\jmath}-3 \mathrm{k})\] Now compare this computed unit vector with the available options: (A) \((1 / \sqrt{7})(\hat{1}+2 \hat{\jmath}-3 \mathrm{k})\) (B) \(-(1 / 2)(\hat{1}+2 \hat{\jmath}-3 \mathrm{k})\) (C) \((1 / \sqrt{14})(\hat{1}+2 \hat{j}-3 \mathrm{k})\) (D) \((1 / \sqrt{5})(\hat{1}+2 \hat{\jmath}-3 \mathrm{k})\) Option (C) matches our computed unit vector, so the correct answer is: (C) \((1 / \sqrt{14})(\hat{1}+2 \hat{j}-3 \mathrm{k})\)

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A freely falling stone crashes through a horizontal glass plate at time \(t\) and losses half of its velocity. After time \((t / 2)\) it falls on the ground. The glass plate is \(60 \mathrm{~m}\) high from the ground. Find the total distance travelled by the stone. \(\left[g=10 \mathrm{~ms}^{-2}\right]\) (A) \(120 \mathrm{~m}\) (B) \(80 \mathrm{~m}\) (C) \(100 \mathrm{~m}\) (D) \(140 \mathrm{~m}\)

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