Question Video: Multiplying Vectors by Scalars Mathematics

Given vectors 𝐀 = ⟨0, 2, −9⟩ and 𝐁 = ⟨1, 3, −4⟩, express 5𝐀 − 3𝐁 in terms of the standard unit vectors 𝐢, 𝐣, and 𝐤.

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Video Transcript

Given vectors 𝐀 equal to zero, two, negative nine and vector 𝐁 equal to one, three, negative four, express five 𝐀 minus three 𝐁 in terms of the standard unit vectors 𝐢, 𝐣, and 𝐤.

In order to answer this question, we firstly need to multiply vector 𝐀 by the scalar five and vector 𝐁 by the scalar three. When multiplying a vector by a scalar, we simply multiply each of the components by that scalar. Five multiplied by vector 𝐀 is therefore equal to five multiplied by the vector zero, two, negative nine. This is equal to zero, 10, negative 45. To find the vector three 𝐁, we multiply the vector one, three, negative four by the scalar three. This is equal to three, nine, negative 12.

Our next step is to subtract these two vectors. We do this by subtracting the corresponding components. Zero minus three is equal to negative three. 10 minus nine is equal to one. Finally, negative 45 minus negative 12 is the same as negative 45 plus 12, which equals negative 33. Five 𝐀 minus three 𝐁 is equal to the vector negative three, one, negative 33.

We are asked to write our answer in terms of the standards unit vectors. We can do this by recalling that the vector with components 𝑥, 𝑦, and 𝑧 can be written as 𝑥 multiplied by 𝐢 hat plus 𝑦 multiplied by 𝐣 hat plus 𝑧 multiplied by 𝐤 hat. The vector negative three, one, negative 33 is therefore equal to negative three 𝐢 plus 𝐣 minus 33𝐤. This is the vector five 𝐀 minus three 𝐁 written in terms of the standard unit vectors.

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