Video Transcript
Two forces 𝐅 sub one equal to two 𝐢 plus three 𝐣 and 𝐅 sub two equal to four 𝐢 plus three 𝐣 are acting on a body. Find the magnitude of the resultant force 𝐑.
We’re given two forces 𝐅 sub one and 𝐅 sub two. And we want to find the magnitude of the resultant force 𝐑. And to do this, we’re going to first need to find the resultant force itself. And we know that for 𝑛 forces 𝐅 sub one, 𝐅 sub two, up to 𝐅 sub 𝑛, the resultant force 𝐑 is 𝐅 sub one plus 𝐅 sub two all the way up to 𝐅 sub 𝑛, that is, the sum of all the forces. In our case, since we have two forces, this means 𝐑 is equal to 𝐅 sub one plus 𝐅 sub two, that is, two 𝐢 plus three 𝐣 plus four 𝐢 plus three 𝐣.
Now, summing vectors in this form, that is, in rectangular form, we add like for like coefficients. And so the coefficient for 𝐢 is now two plus four, and the coefficient of 𝐣 is now three plus three, so that our resultant force 𝐑 is six 𝐢 plus six 𝐣.
Now, for a vector 𝐕 equal to 𝑥𝐢 plus 𝑦𝐣, which is in two-dimensional rectangular form, its magnitude is given by the square root of the sum of the squares of the coefficients of 𝐢 and 𝐣, that is, the square root of 𝑥 squared plus 𝑦 squared. For our resultant force 𝐑, both 𝑥 and 𝑦 are six. So its magnitude is the square root of six squared plus six squared. We can write this as the square root of two times six squared. And since the square root of six squared is simply six, this is equal to six times the square root of two.
And so for the two forces 𝐅 sub one and 𝐅 sub two, the magnitude of the resultant force 𝐑 is six times the square root of two.
