Find the resultant of two vectors, 3 unit and 4 unit acting at point O at an angle 45° with each other.

Steps to Solve

1. Identify the vectors and angle The two vectors are given:

• Magnitude of the first vector $A = 3$ units
• Magnitude of the second vector $B = 4$ units The angle between them is $θ = 45^\circ$.

2. Use the Law of Cosines to find the resultant The magnitude of the resultant vector $R$ can be found using the formula: $$ R = \sqrt{A^2 + B^2 + 2AB \cos(θ) } $$ Substituting the values: $$ R = \sqrt{3^2 + 4^2 + 2 \cdot 3 \cdot 4 \cdot \cos(45^\circ)} $$

3. Calculate cosine and substitute Knowing that $\cos(45^\circ) = \frac{\sqrt{2}}{2}$, rewrite the equation: $$ R = \sqrt{9 + 16 + 24 \cdot \frac{\sqrt{2}}{2}} $$ Simplify: $$ R = \sqrt{25 + 12\sqrt{2}} $$

4. Find the direction of the resultant To find the angle $\phi$ of the resultant with respect to vector A, use the Law of Sines: $$ \frac{R}{\sin(θ)} = \frac{B}{\sin(\phi)} $$ Rearranging gives: $$ \sin(\phi) = \frac{B \sin(θ)}{R} $$

5. Calculate the angle Substituting in values: $$ \sin(\phi) = \frac{4 \cdot \sin(45^\circ)}{R} $$ Knowing that $\sin(45^\circ) = \frac{\sqrt{2}}{2}$, substitute into the equation.