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Chapter 17: Problem 1648

the angle between \(\underline{a}\) and \(\underline{b}\) is \((5 \pi / 6)\), projectile of a over \(\underline{b}\) is \((6 / \sqrt{3})\) then the value \(|\underline{\mathrm{a}}|=\underline{(\underline{\mathrm{a}}, \underline{\mathrm{b}}, \neq \underline{0})}\) (a) 12 (b) 6 (c) 4 (d) \((\sqrt{3} / 2)\)

Short Answer

Expert verified
The magnitude of vector \(\underline{a}\) is 4 (c).

Step by step solution

01

Formula for projection of a vector

The projection of vector \(\underline{a}\) over \(\underline{b}\) can be found using the formula: \[proj_{\underline{b}}(\underline{a}) = \frac{\underline{a} \cdot \underline{b}}{|\underline{b}|}\]
02

Angle provided, use projection-appraoch

Since we are given the angle between the vectors, it is easier to use the approach: \[proj_{\underline{b}}(\underline{a}) = |\underline{a}|\cos(\theta)\] We know that the angle between \(\underline{a}\) and \(\underline{b}\) is \((5 \pi / 6)\) and the projection of \(\underline{a}\) over \(\underline{b}\) is \((6 / \sqrt{3})\). We can plug these values into the equation.
03

Solving for the magnitude of vector \(\underline{a}\)

With the given values of \(\theta\) and the projectile of \(\underline{a}\) over \(\underline{b}\), we can solve for the magnitude of vector \(\underline{a}\). \[ \frac{6}{\sqrt{3}} = |\underline{a}|\cos\left(\frac{5 \pi}{6}\right) \] Since \(\cos(\frac{5\pi}{6}) = -\frac{\sqrt{3}}{2}\), the equation becomes: \[ \frac{6}{\sqrt{3}} = |\underline{a}|\left(-\frac{\sqrt{3}}{2}\right) \]
04

Isolate |\(\underline{a}\)| and calculate its value

Now, we need to isolate \(|\underline{a}|\) and find its value. Divide both sides of the equation by \((-\frac{\sqrt{3}}{2})\): \[ |\underline{a}| = \frac{6}{\sqrt{3}}\times \frac{-2}{\sqrt{3}} \] \[ |\underline{a}| = -\frac{12}{3} \] Since the magnitude of a vector is always non-negative, we consider only the positive value of the expression: \[ |\underline{a}| = 4 \] So, the magnitude of vector \(\underline{a}\) is 4, which corresponds to the answer (c).

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Most popular questions from this chapter

\(\underline{\mathrm{a}}, \underline{\mathrm{b}}\) and \(\underline{\text { c }}\) are non zero vectors, if \(\underline{\mathrm{a}}=8 \underline{\mathrm{b}}\) and \(\underline{\mathrm{c}}=-7 \underline{\mathrm{b}}\) then the angle between \(\underline{a}\) and \(\underline{c}\) is (a) \(\pi\) (b) 0 (c) \((\pi / 4)\) \((\mathrm{d})(\pi / 2)\)

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.

find the number of vectors in \(\mathrm{R}^{3}\) such the angle between X-axis and vectors are \((\pi / 3)\) (a) 1 (b) 2 (c) 4 (d) infinite times

vector \(\underline{b}=(0,3,4)\) is represented by \(\underline{b}_{1}\) and \(\underline{b}_{2}\) where \(\underline{b}_{1}\) is same direction of \(\underline{a}=(1,1,0)\) and \(\underline{b}_{2}\) is perpendicular, then \(\underline{\mathrm{b}}_{2}=\) (a) \([(3 / 2),(3 / 2), 0]\) (b) \([(3 / 2),\\{(-3) / 2\\}, 4]\) (c) \([0,(3 / 5),(4 / 5)]\) (d) none of these

\(\underline{\mathrm{a}}=(1,-1,0), \underline{\mathrm{b}}=(0,1,-1)\) and \(\underline{\mathrm{c}}=(-1,0,1)\) find the unit vector \(\underline{d}\) Such that \(\underline{\mathrm{a}} \cdot \underline{\mathrm{d}}=0=[\underline{\mathrm{b}} \underline{\mathrm{c}} \underline{\mathrm{d}}]\). (a) \(\pm(1 / \sqrt{6})(1,1,-2)\) (b) \(\pm(1 / \sqrt{3})(1,1,-1)\) (c) \(\pm(1 / \sqrt{3})(1,1,1)\) (d) \(\pm(0,0,1)\)
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