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

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

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If 𝐀 = <−8, 9, 9> and 𝐁 = <−6, 4, 9>, find (2/5 𝐀) − (4/5 𝐁).

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Video Transcript

If vector 𝐀 is equal to negative eight, nine, nine and vector 𝐁 is equal to negative six, four, nine, find two-fifths of vector 𝐀 minus four-fifths of vector 𝐁.

In order to answer this question, we will begin by multiplying vector 𝐀 by the scalar two-fifths. We will then multiply vector 𝐁 by the scalar four-fifths. We will then subtract the two vectors to calculate two-fifths of vector 𝐀 minus four-fifths of vector 𝐁.

We recall that in order to multiply any vector by a scalar, we simply multiply each of the individual components by that scalar or constant. Two-fifths multiplied by negative eight is negative sixteen-fifths. And two-fifths multiplied by nine is equal to eighteen-fifths. Two-fifths of vector 𝐀 is equal to negative sixteen-fifths, eighteen-fifths, eighteen-fifths.

We can use a similar method to calculate four-fifths of vector 𝐁. Four-fifths multiplied by negative six is negative twenty-four fifths. Four-fifths multiplied by four is equal to sixteen-fifths. Finally, four-fifths multiplied by nine is equal to thirty-six fifths. Four-fifths of vector 𝐁 is therefore equal to negative twenty-four fifths, sixteen-fifths, thirty-six fifths.

We can now subtract these two vectors by subtracting their corresponding components. Negative sixteen-fifths minus negative twenty-four fifths is the same as negative sixteen fifths plus twenty-four fifths. This is equal to eight-fifths. Eighteen-fifths minus sixteen-fifths is equal to two-fifths. Finally, eighteen-fifths minus thirty-six fifths is equal to negative eighteen-fifths.

If vector 𝐀 is equal to negative eight, nine, nine and vector 𝐁 is equal to negative six, four, nine, then two-fifths of vector 𝐀 minus four-fifths of vector 𝐁 is equal to eight-fifths, two-fifths, negative eighteen-fifths.

An alternative method here to avoid working with fractions all the way through would be to factor out either one-fifths or two-fifths from our initial expression. For example, two-fifths of vector 𝐀 minus four-fifths of vector 𝐁 is the same as one-fifth multiplied by two times vector 𝐀 minus four times vector 𝐁. We could then calculate two times vector 𝐀 and four times vector 𝐁 before subtracting these vectors and then multiplying the answer by the scalar one-fifth.

Two 𝐀 minus four 𝐁 gives us the vector eight, two, negative 18. Multiplying this by the scalar one-fifth once again gives us the vector eight-fifths, two-fifths, negative eighteen-fifths.

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