Vector resolution involving angles
When two vectors are inclined at an angle to each other, they cannot be resolved by simple addition and
subtraction. The resolution of these vectors is governed by the parallelogram and triangular law of vectors.
Parallelogram law of vectors
The parellogram law of vectors states that if two vectors are represented in magnitude and direction by the
adjacent sides of a parallelogram, the resultant vector is represented in magnitude and direction as the
diagonal that cuts across the two vectors.
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Triangular law of vectors
Triangle law of vectors states that when two vectors are represented as two sides of the triangle with the
order of magnitude and direction, then the third side of the triangle represents the magnitude and direction
of
the resultant vector.
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Lami's theory of vectors
Lami's Theory states that if three coplanar forces acting on a point are in equilibrium, then each force is proportional to the sine of the angle between the other two forces. Mathematically, this can be expressed as:
$$ \frac{a}{sinA} = \frac{b}{sinB} = \frac{c}{sinC} $$
where a, b, and c are the magnitudes of the forces, and A, B, and C are the angles between the forces.
Deductions
For two vectors P and Q inclined at right angles or 90° to each other,
the resultant vector, R and direction \( \alpha \) is given by:
$$ R = \sqrt{P² + Q²} $$
where P is the vertical component(opposite) and Q is the horizontal component(adjacent).
$$ \alpha = tan^{-1}(\frac{P}{Q})$$
For two vectors inclined at an angle less than or greater than 90°, the resultant R and direction \( \alpha \) is given by:
$$ R = \sqrt{P²+Q²+2PQCos\theta} $$
$$ \alpha = tan^{-1}(\frac{Psin\theta}{Q + Pcos\theta}) $$