Question Video: Understanding Unit Vectors in Relation To Scalar Multiplication Mathematics

Consider the vector 𝐀 = 5𝐒 βˆ’ 2𝐣 βˆ’ 4𝐀. Is the unit vector in the direction of 𝐀 the same as the unit vector in the direction of 3𝐀?

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Video Transcript

Consider the vector 𝐀 five 𝐒 minus two 𝐣 minus four 𝐀. Is the unit vector in the direction of 𝐀 the same as the unit vector in the direction of three 𝐀?

We know that vector 𝐀 can be rewritten in the form five, negative two, negative four. We are also asked to consider the vector three 𝐀. This involves multiplying vector 𝐀 by the scalar or constant three. This involves multiplying each component by the scalar three. Three multiplied by five is 15. Three multiplied by negative two is negative six. And three multiplied by negative four is negative 12. This means that three 𝐀 is equal to 15, negative six, negative 12.

The unit vector 𝐕 hat is equal to one over the magnitude of vector 𝐕 multiplied by vector 𝐕. This is the same as dividing the vector by its magnitude. We know that to calculate the magnitude of any vector, we find the square root of the sum of the squares of each component. The magnitude of vector 𝐀 is equal to the square root of five squared plus negative two squared plus negative four squared. This is equal to the square root of 25 plus four plus 16. This simplifies to the square root of 45, which in turn is equal to three root five.

The magnitude of vector 𝐀 is three root five. The unit vector in the direction of vector 𝐀 is therefore equal to one over three root five multiplied by five, negative two, negative four. We could rewrite this as the vector five over three root five, negative two over three root five, and negative four over three root five.

Let’s now consider the unit vector in the direction of three 𝐀. The magnitude of vector three 𝐀 is equal to the square root of 15 squared plus negative six squared plus negative 12 squared. This is equal to the square root of 405, which in turn simplifies to nine root five. The magnitude of vector three 𝐀 is equal to nine root five.

We might notice at this point that this is three times the magnitude of vector 𝐀. This leads us to a general rule. The magnitude of π‘˜π• is equal to π‘˜ multiplied by the magnitude of vector 𝐕. This means that three multiplied by the magnitude of vector 𝐀 is equal to three multiplied by three root five. The unit vector in the direction of three 𝐀 is therefore equal to one over nine root five multiplied by the vector 15, negative six, negative 12.

Factoring out a three from the vector gives us three over nine root five multiplied by the vector five, negative two, negative four. This is the same as one over three root five multiplied by the vector five, negative two, negative four. We can therefore conclude that the answer is yes, the unit vector in the direction of 𝐀 is the same as the unit vector in the direction of three 𝐀.

This will be true of any vector multiplied by a positive scalar. As long as our value of π‘˜ is positive, then the unit vector in the direction of 𝐀 will be the same as the unit vector in the direction of π‘˜π€.

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