Question Video: Solving a Vector Equation Mathematics

Video Transcript

Given that 𝐀 is the vector one, two, two and 𝐁 is the vector four, three, one, find the vector 𝐂 that satisfies the equation 𝐂 is equal to two 𝐀 plus 𝐁.

In this question, we’re given two three-dimensional vectors, the vector 𝐀 and the vector 𝐁. We’re given the components of these two vectors. We need to use these to determine the vector 𝐂 which is equal to two 𝐀 plus 𝐁. Since we’re told that vector 𝐂 is equal to two times vector 𝐀 plus vector 𝐁, we can start by substituting vector 𝐀 and vector 𝐁 into this expression. This gives us that vector 𝐂 is equal to two times the vector one, two, two added to the vector four, three, one. We now need to evaluate the right-hand side of this expression. First, we have two multiplied by a vector. And remember, when we multiply a scalar by a vector, we need to multiply all of the components by this scalar.

So in this case, we need to multiply all of the components of vector 𝐀 by the scalar of two. This gives us the vector two times one, two times two, two times two. And then after this, we still need to add our vector 𝐁. Next, we can evaluate all of the expressions for the components of the vector two 𝐀. This gives us the vector two, four, four, and we need to add on to this the vector four, three, one. Remember, to add two vectors of the same dimension together, we just need to add the corresponding components together. Applying this to our two vectors, we get the vector two plus four, four plus three, four plus one. And finally, since two plus four is equal to six, four plus three is equal to seven, and four plus one is equal to five, this simplifies to give us the vector six, seven, five, which is our final answer.

Therefore, given that the vector 𝐀 is one, two, two and the vector 𝐁 is four, three, one, then the vector 𝐂 which satisfies the equation 𝐂 is equal to two 𝐀 plus 𝐁 will be the vector six, seven, five.