Question Video: Determining the Relations between Two Vectors Mathematics

Given the two vectors 𝐀 = (8𝐒 βˆ’ 7𝐣 + 𝐀) and 𝐁 = (64𝐒 βˆ’ 56𝐣 + 8𝐀), determine whether these two vectors are parallel, perpendicular, or otherwise.

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Video Transcript

Given the two vectors 𝐀 is equal to eight 𝐒 minus seven 𝐣 plus 𝐀 and 𝐁 is equal to 64𝐒 minus 56𝐣 plus eight 𝐀, determine whether these two vectors are parallel, perpendicular, or otherwise.

We know that two vectors are parallel if they are scalar multiples of each other. Vector 𝐀 must be equal to π‘˜ multiplied by vector 𝐁. Two vectors 𝐀 and 𝐁 are perpendicular on the other hand, if their scalar or dot product is equal to zero. Let’s firstly consider whether our two vectors 𝐀 and 𝐁 are parallel. If one vector is a scalar multiple of another vector, then the ratio of their individual components must be equal. In this case, 64 over eight must be equal to negative 56 over negative seven, which must be equal to eight over one. 64 divided by eight is equal to eight, and eight divided by one is equal to eight.

Dividing a negative number by a negative number gives a positive answer. Therefore, negative 56 divided by negative seven is also equal to eight. We can therefore conclude that vector 𝐁 is equal to eight multiplied by vector 𝐀 or vector 𝐀 is equal to one-eighth of vector 𝐁. The two vectors 𝐀 and 𝐁 are therefore parallel. Whilst they cannot be parallel and perpendicular, let’s just check that the scalar product is not equal to zero. The 𝐒-components of our vector are eight and 64. The 𝐣-components are negative seven and negative 56. The 𝐀-components are one and eight. Eight multiplied by 64 is 512. Negative seven multiplied by negative 56 is 392. One multiplied by eight is equal to eight. The scalar product of vectors 𝐀 and 𝐁 is therefore equal to 912. As this is not equal to zero, the vectors 𝐀 and 𝐁 are not perpendicular.

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