First slide

Vectors

Question

The resultant of three vectors 1, 2 and 3 units whose directions are those of the sides of an equilateral triangle is

Moderate

A

at an angle of 30° with the first vector
B

at an angle of 15° with the first vector
C

at an angle of 100° with the first vector
D

at an angle of 150° with the first vector

Solution

The vectors are shown in figure
Resolving the vectors along X, Y and Z axes, we have
R_x =1+2cos120°+3cos240°
R_y = 2 sin 120° + 3 sin 240°
$\begin{array}{l} ∴ R_{x} = 1 - 2 \times \frac{1}{2} - 3 \times \frac{1}{2} = - \frac{3}{2} \\ and R_{y} = 2 \times (\sqrt{3} / 2) - 3 \times (\sqrt{3} / 2) = - \sqrt{3} / 2 \\ Now \tan θ = \frac{R_{y}}{R_{x}} = - \frac{(\sqrt{3} / 2)}{- (3 / 2)} = \frac{1}{\sqrt{3}} \\ θ = \tan^{- 1} (1 / \sqrt{3}) = 180^{0} - 30^{\circ} = 150^{0} \end{array}$

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