Given the vectors 𝐀 is equal to two, three, one and 𝐁 is equal to six, nine, three, write vector 𝐁 as a scalar multiple of vector 𝐀 if possible. Is it option (A) 𝐁 is equal to one-third 𝐀, option (B) 𝐁 is equal to one-half 𝐀, option (C) 𝐁 is equal to three 𝐀, option (D) 𝐁 is equal to six 𝐀, or option (E) it is not possible?
We are asked to try and write in this question vector 𝐁 as a scalar multiple of vector 𝐀. This means that 𝐁 must be equal to 𝑘 multiplied by 𝐀. We are told that vector 𝐁 is equal to six, nine, three. This is equal to the constant or scalar 𝑘 multiplied by the vector two, three, one. When multiplying any vector by a scalar, we multiply each of the components by that scalar. 𝑘 multiplied by the vector two, three, one is equal to the vector two 𝑘, three 𝑘, 𝑘.
We can now compare the 𝑥-, 𝑦-, and 𝑧-components on each side of our equation. Firstly, we have six is equal to two 𝑘. Dividing both sides of this equation by two gives us 𝑘 is equal to three. When we compare the 𝑦-components, we have nine is equal to three 𝑘. Dividing both sides of this equation by three once again gives us an answer of 𝑘 is equal to three. Finally, considering the 𝑧-components, we have three is equal to 𝑘 or 𝑘 is equal to three. As our value of 𝑘 is the same for all three components, vector 𝐁 can be written as a scalar multiple of vector 𝐀. The correct answer is option (C): vector 𝐁 is equal to three multiplied by vector 𝐀.