Question

Given vectors u and v such that the angle between the two vectors is 30° and their magnitudes are 5 and
8 respectively, determine vector u + vector v.​

Answer 1

Understanding the question -

=> Here the concept of Vector addition is used.

=> We use the Triangle Rule of Vector addition in this case.

Points to note -

=> The angle between two vectors is always measured when the two vectors are joined at their tails .

=> The Triangle rule of vector addition is used when the the head of the first vector touches the tail of the second vector.

Given -

=> Magnitude of 1st vector $|\overrightarrow{u}|$ = 5 units

=> Magnitude of 2nd vector $|\overrightarrow{v}|$ = 8 units

=> θ (Angle between the two vectors when joined at their tails) = 30 °

Diagram -

=> I have attached the diagram below .

=> Here, the resultant is calculated by the Triangle rule of vector addition .

Calculation -

=> We will use the formula -

$\boxed{R\:(Resultant) = \sqrt{u^{2} + v^{2} + 2uv*cos\:\theta } }$

➼ $\large{R = \sqrt{5^{2} +8^{2} + 2 * 5*8*cos 30 \textdegree } }$

[ cos 30° = $\dfrac{\sqrt{3} }{2}$ ]

➼ R = $\sqrt{25+64+2*40*\dfrac{\sqrt{3} }{2} }$

➼ R = $\sqrt{25+64+40\sqrt{3} }$

➼ $R =\sqrt{89 + 40\sqrt{3} }$

Answer -

=> The Resultant or $|\overrightarrow{u}| + |\overrightarrow{v}|$ will be $\sqrt{89 + 40\sqrt{3} }$ units .

More to know -

=> For the parallelogram rule of vector addition, the two vectors must be joined at their tails.

=> In the parallelogram rule of vector addition, the resultant of the two vectors is given by the Diagonal of the parallelogram which passes through the point of attachment of the tails of the two vectors.

=> Refer to attachment 2 for more information about the Parallelogram rule of vector addition.