Question Video: Evaluating Expressions Involving the Subtraction and Scalar Multiplication of Given Vectors in Three Dimensions | Nagwa Question Video: Evaluating Expressions Involving the Subtraction and Scalar Multiplication of Given Vectors in Three Dimensions | Nagwa

Question Video: Evaluating Expressions Involving the Subtraction and Scalar Multiplication of Given Vectors in Three Dimensions Mathematics • Third Year of Secondary School

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If 𝐀 = 9𝐒 βˆ’ 7𝐣 + 8𝐀 and 𝐁 = 8𝐒 βˆ’ 3𝐣 + 4𝐀, find 𝐀 βˆ’ (2/3 𝐁).

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Video Transcript

If vector 𝐀 is equal to nine 𝐒 minus seven 𝐣 plus eight 𝐀 and vector 𝐁 is equal to eight 𝐒 minus three 𝐣 plus four 𝐀, find vector 𝐀 minus two-thirds of vector 𝐁.

In this question, we will need to recall how to subtract vectors and also how to multiply a vector by a scalar. Let’s begin by calculating two-thirds of vector 𝐁.

In order to multiply any vector by a scalar, we simply multiply each of the components by that scalar. Two-thirds multiplied by eight is equal to sixteen-thirds. Therefore, two-thirds multiplied by eight 𝐒 is sixteen-thirds 𝐒. Multiplying two-thirds by negative three 𝐣 gives us negative two 𝐣. Finally, multiplying two-thirds by four 𝐀 gives us eight-thirds 𝐀. Multiplying vector 𝐁 by the scalar two-thirds gives us sixteen-thirds 𝐒 minus two 𝐣 plus eight-thirds 𝐀.

We need to subtract this vector from vector 𝐀. In order to subtract two vectors, we simply subtract the corresponding components. Nine 𝐒 minus sixteen-thirds 𝐒 is equal to eleven-thirds 𝐒. This is because nine can be rewritten as twenty-seven thirds or 27 over three. Subtracting the 𝐣-components gives us negative five 𝐣, as negative seven 𝐣 minus negative two 𝐣 is the same as negative seven 𝐣 plus two 𝐣.

Finally, eight 𝐀 minus eight-thirds 𝐀 is equal to sixteen-thirds 𝐀. We know this as we can rewrite eight as twenty-four thirds. And twenty-four thirds minus eight-thirds is equal to sixteen-thirds. Vector 𝐀 minus two-thirds of vector 𝐁 is therefore equal to eleven-thirds 𝐒 minus five 𝐣 plus sixteen-thirds 𝐀.

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