Question Video: Finding the Work Done by a Force in Vector Form Acting on a Body Moving between Two Points Mathematics

Video Transcript

A particle moved from point 𝐴 negative two, negative two to point 𝐵 six, 10 along a straight line under the action of the force 𝐅 equal to 𝑘𝐢 minus six 𝐣 acting in the opposite direction to the displacement 𝐀𝐁. Find the work done by the force 𝐅.

We begin by recalling that the work done by a force can be calculated by finding the dot product of the vector force 𝐅 and displacement force 𝐝. In this question, we are told that 𝐅 is equal to 𝑘𝐢 minus six 𝐣 and that it moves from point 𝐴 with coordinates negative two, negative two to point 𝐵 with coordinates six, 10.

The displacement 𝐝 is therefore equal to the vector 𝐀𝐁. And we can calculate this by subtracting the position vector of point 𝐴 from the position vector of point 𝐵. We have six 𝐢 plus 10𝐣 minus negative two 𝐢 minus two 𝐣. And this is equal to eight 𝐢 plus 12𝐣. The work done will be equal to the scalar product of the two vectors 𝑘𝐢 minus six 𝐣 and eight 𝐢 plus 12𝐣. However, at present, we don′t know the value of the constant 𝑘.

We are told, however, that the force 𝐅 acts in the opposite direction to the displacement 𝐀𝐁. This means that it must travel in the same direction as the vector 𝐁𝐀, which is equal to negative eight 𝐢 minus 12𝐣. Comparing the 𝐢- and 𝐣-components of this vector with vector 𝐅, we notice that the 𝐣-component has been multiplied by the scalar two. In order for 𝐅 to have the same direction as 𝐁𝐀, the same must be true of the 𝐢-components. 𝑘 multiplied by two or two 𝑘 must be equal to negative eight.

Dividing through by two, we have 𝑘 is equal to negative four. And this means that the force vector is equal to negative four 𝐢 minus six 𝐣. Clearing some space, we can now calculate the work done. It is equal to the scalar product of negative four 𝐢 minus six 𝐣 and eight 𝐢 plus 12𝐣. The scalar product is equal to the sum of the products of the individual components, giving us negative four multiplied by eight plus negative six multiplied by 12. This is equal to negative 32 minus 72, which gives us a final answer of negative 104.

Since there are no units given in this question, we can conclude that the work done by the force 𝐅 is negative 104 units of work.