Video Transcript
Given that vector 𝐀 equals one, zero, negative one and vector 𝐂 equals three, one, four, determine four multiplied by vector 𝐎 minus two multiplied by vector 𝐂 plus four multiplied by vector 𝐀, where vector 𝐎 is a zero vector.
We know that the zero vector must have all components equal to zero. And when performing these operations, each vector must have the same number of components. Therefore, the zero vector in this question is equal to zero, zero, zero. When multiplying any vector by a scalar, we simply multiply each of the components by that scalar. This means that four multiplied by the zero vector is equal to zero, zero, zero. Four multiplied by vector 𝐀 is equal to four, zero, negative four.
Inside the parentheses, we need to add vector 𝐂 to four 𝐀. We can add two vectors by adding their corresponding components. Three plus four is equal to seven. One plus zero is equal to one. And four plus negative four is equal to zero. Therefore, vector 𝐂 plus four multiplied by vector 𝐀 is equal to seven, one, zero.
Four multiplied by the zero vector minus two multiplied by vector 𝐂 plus four multiplied by vector 𝐀 is therefore equal to zero, zero, zero minus two multiplied by seven, one, zero. Once again, we can multiply a vector by a scalar by multiplying each of the components by this scalar. This gives us zero, zero, zero minus 14, two, zero. We can then subtract each of the corresponding components, giving us negative 14, negative two, zero.
Four multiplied by the zero vector minus two multiplied by 𝐂 plus four 𝐀 is equal to negative 14, negative two, zero.