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Chapter 17: Problem 1667

if the angle between \(\underline{a}\) and \(\underline{b}\) is \(\theta\) then \([(|\underline{a} \times \underline{b}|) /(\underline{a} \cdot \underline{b})]=\) (a) \(\tan \theta\) (b) \(-\tan \theta\) (c) \(\cot \theta\) (d) \(-\cot \theta\)

Short Answer

Expert verified
The short answer based on the step-by-step solution is: \[\frac{|\underline{a} \times \underline{b}|}{\underline{a} \cdot \underline{b}} = \tan{\theta}\]

Step by step solution

01

Write the formulas for cross product and dot product magnitude

We know that the cross product of two vectors \(\underline{a}\) and \(\underline{b}\) is given by \(|\underline{a} \times \underline{b}| = |\underline{a}| |\underline{b}| \sin{\theta}\) and their dot product is given by \(\underline{a} \cdot \underline{b} = |\underline{a}| |\underline{b}| \cos{\theta}\), where \(|\underline{a}|\) and \(|\underline{b}|\) represent the magnitudes of the vectors and \(\theta\) is the angle between them.
02

Calculate the ratio asked in the problem

Divide the magnitude of the cross product by the dot product: \[\frac{|\underline{a} \times \underline{b}|}{\underline{a} \cdot \underline{b}} = \frac{|\underline{a}| |\underline{b}| \sin{\theta}}{|\underline{a}| |\underline{b}| \cos{\theta}}\]
03

Simplify the ratio

Since \(|\underline{a}|\) and \(|\underline{b}|\) are common in both numerator and denominator, they can be cancelled out: \[\frac{|\underline{a} \times \underline{b}|}{\underline{a} \cdot \underline{b}} = \frac{\sin{\theta}}{\cos{\theta}}\]
04

Identify the expression for the ratio

The expression \(\frac{\sin{\theta}}{\cos{\theta}}\) is the definition of the tangent function of \(\theta\), i.e., \(\tan{\theta}\). So, the correct answer is: (a) \(\tan \theta\)

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Most popular questions from this chapter

if the vectors \(10 \underline{i}+3 j, 12 \dot{i}-5 j\) and ai \(+11 j\) are collinear then \(\mathrm{a}=\) (a) \(\overline{-8}\) (b) 4 (c) 8 (d) 12

if \(\underline{a}, \underline{b}\) and \(\underline{c}\) such a vectors that \(\underline{a} \neq \underline{0}, \underline{a} \times \underline{b}=2 \underline{a} \times \underline{c}\), \(|\underline{a}|=|\underline{c}|=1,|\underline{b}|=4\) and \(|\underline{b} \times \underline{c}|=\sqrt{1} 5\) if \(\underline{b}-2 \underline{c}=\lambda \underline{a}\) then \(\lambda=\) (a) \(-1\) (b) 1 (c) 2 (d) \(\pm 4\)

if the vectors \(\underline{a}\) and \(\underline{b}\) such that \(|\underline{\mathrm{a}}+\underline{\mathrm{b}}|<|\underline{\mathrm{a}}-\underline{\mathrm{b}}|\) the angle between \(\underline{a}\) and \(\underline{b}\) is. (a) acute (b) right angle (c) obtuse (d) none of these.

if \(\underline{\mathrm{a}}=\underline{\mathrm{i}}-\mathrm{j}+2 \underline{\mathrm{k}}, \underline{\mathrm{b}}=2 \underline{\mathrm{i}}+4 \mathrm{j}+4 \underline{\mathrm{k}}\) and \(\underline{\mathrm{c}}=\lambda_{\underline{\mathrm{i}}}+\mathrm{j}+\mu \mathrm{k}\) are perpendicular then the value of \((\lambda, \mu)=\) (a) \((-3,2)\) (b) \((2,-3)\) (c) \((-2,3)\) (d) \((3,-2)\)

\((1 / \mathrm{c}),(1 / \mathrm{c}),(1 / \mathrm{c})\) is direction cosine of the line then the value of \(\mathrm{c}=\) (a) \(\pm(\overline{1 / 3)}\) (b) \(\pm 3\) (c) \(\pm(1 / \sqrt{3})\) (d) \(\pm \sqrt{3}\)
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