The vector P makes 120° with the x-axis and the vector Q makes 30° with the y-
axis. What is their resultant?
(a) P+Q
(b)P-O
(c) /P²+Q²
(d) /p² – Q²
Answers
Given info : The vector P makes 120° with the x-axis and the vector Q makes 30° with the y- axis.
To find : the resultant of vectors P and Q.
Solution :
Method 1 : draw each of them, you will get angle between P and Q is 0° [ see figure ]
So, resultant, R = √{P² + Q² + 2PQcos0°}
= |P + Q|
Therefore the resultant is |P + Q|
method 2 : as P makes an angle 120° with x - axis.
In vector form P = Pcos120° i + Psin120° j
Similarly, Q makes 30° with y - axis.
In vector form Q = Qsin30° i + Qcos30° j
Now resultant of P and Q , R = P + Q
= (Pcos120° i + Psin120° j) + (-Qsin30° i + Qcos30° j)
= (P × -1/2 - Q × 1/2) j + (P × √3/2 + Q × √3/2) j
= 1/2(-P - Q)i + √3/2 (P + Q) j
Now magnitude |R| = √[1/4(-P - Q)² + 3/4(P + Q)²}
= √{P² + Q² + 2PQ}
= P + Q
Therefore the resultant must be |P + Q|