The angle between the vectors \(\overrightarrow {A} = \hat{i} + \hat{j}\) and \(\overrightarrow {B} = \hat{i} + \hat{j} + c\hat {k}\) is 30o.Find the unknown \(c\) .
The angle between the vectors \(\overrightarrow {A} = \hat{i} + \hat{j}\) and \(\overrightarrow {B} = \hat{i} + \hat{j} + c\hat {k}\) is 30o.Find the unknown \(c\) .
- zero
1
- $ \pm \sqrt{\frac{2}{3}} $
- $ \pm \frac{1}{2} $
The Correct Option is C
Solution and Explanation
\(A = i + j, B = i + j + c\hat{k}\)
\(A\cdot B = AB\,cos\,\theta\)
\(( i + j) \cdot (i + j + ck) = \sqrt{2} \times \sqrt{2 + c^2} \times \frac{\sqrt{3}}{2}\)
\((\because \theta = 30^{\circ})\)
\(2 = \sqrt{2} \sqrt{2+ c^2} \times \frac{\sqrt{3}}{2}\)
\(4 = \sqrt{12 + 6c^2}\)
or \(16 = 12 + 6c^2\)
\(c^2 = \frac{4}{6} = \frac{2}{3}\)
\(c = \pm \sqrt{\frac{2}{3}}\)
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Concepts Used:
Work
Work is the product of the component of the force in the direction of the displacement and the magnitude of this displacement.
Work Formula:
W = Force × Distance
Where,
Work (W) is equal to the force (f) time the distance.
Work Equations:
W = F d Cos θ
Where,
W = Amount of work, F = Vector of force, D = Magnitude of displacement, and θ = Angle between the vector of force and vector of displacement.
Unit of Work:
The SI unit for the work is the joule (J), and it is defined as the work done by a force of 1 Newton in moving an object for a distance of one unit meter in the direction of the force.
Work formula is used to measure the amount of work done, force, or displacement in any maths or real-life problem. It is written as in Newton meter or Nm.
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