### Video Transcript

A force of negative 𝐢 plus 𝐣 plus 𝐤 newtons is being applied to an object. What other force should be applied to achieve a total force of two 𝐢 plus 𝐣 plus 𝐤 newtons?

We are told in the question that a force of negative 𝐢 plus 𝐣 plus 𝐤 newtons is being applied to an object. Written in component form, this force is equal to negative one, one, one. A second force is also applied to the object. It is this force we are trying to calculate. And we will let it have components 𝐱, 𝐲, and 𝐳. These two forces have a total force of two 𝐢 plus 𝐣 plus 𝐤, which in component form is two, one, one.

Since the resultant force is the sum of the other two vector forces, our three vectors satisfy the equation 𝐱, 𝐲, 𝐳 plus negative one, one, one is equal to two, one, one. Subtracting the vector negative one, one, one from both sides, we have 𝐱, 𝐲, 𝐳 is equal to two, one, one minus negative one, one, one.

We recall that when subtracting two vectors, we simply subtract their corresponding components. Two minus negative one is the same as two plus one, which is equal to three. Both the 𝐲- and 𝐳-components on the right-hand side are equal to zero as one minus one is zero. The vector 𝐱, 𝐲, 𝐳 is equal to three, zero, zero. As both 𝐲 and 𝐳 are zero, there is no 𝐣- or 𝐤-component to our vector.

The additional force that needs to be applied to the object to achieve a total force of two 𝐢 plus 𝐣 plus 𝐤 newtons is equal to three 𝐢 newtons.