the resultant of two vectors P and Q is R. if Q is doubled, the new resultant is perpendicular to P. Then R equals

(a) P

(b) (P+Q)

( c) Q

(d) (P-Q)

please explain how to solve

To tackle the problem involving vectors P and Q, we first need to understand the relationships between these vectors and their resultant. We know that the resultant of vectors P and Q is R, so we can express this mathematically as:

R = P + Q

Now, the problem states that if vector Q is doubled, the new resultant becomes perpendicular to vector P. Let's denote the new vector as Q' where:

Q' = 2Q

With this change, the new resultant R' can be expressed as:

R' = P + Q' = P + 2Q

For R' to be perpendicular to P, the dot product of R' and P must equal zero:

R' • P = 0

Substituting R' into the dot product condition gives us:

(P + 2Q) • P = 0

Expanding the dot product results in:

P • P + 2Q • P = 0

Here, P • P is simply the magnitude of vector P squared (let's denote it as |P|²), and Q • P is the dot product of vectors Q and P. This leads us to:

|P|² + 2(Q • P) = 0

From this equation, we can isolate Q • P:

2(Q • P) = -|P|²

Thus, we find:

Q • P = -\frac{|P|²}{2}

Next, we need to evaluate what R equals based on our original options. Initially, we expressed R as:

R = P + Q

Now, we can analyze the four options provided:

• (a) R = P
• (b) R = P + Q
• (c) R = Q
• (d) R = P - Q

Given that R = P + Q, we see that option (b) matches our expression for the resultant vector R. Since the problem states that the new resultant R' is perpendicular to P when Q is doubled, this condition does not alter the original relationship of R being P + Q.

Therefore, the correct answer is:

(b) R = P + Q

In summary, by establishing the relationship between the vectors, applying the condition for perpendicularity, and checking the options, we determined that the resultant vector R retains its original form as P + Q.