Question Video: Combining Scalar Multiplication, Vector Operations and Unit Vector Calculations Mathematics

Given 𝐀 = 〈2, 0, −2〉 and 𝐁 = 〈1, −1, 1〉, determine the unit vector in the direction 2𝐁 − 𝐀.

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Video Transcript

Given vector 𝐀 is equal to two, zero, negative two and vector 𝐁 is equal to one, negative one, one, determine the unit vector in the direction two 𝐁 minus 𝐀.

Our first step here is to calculate the vector two 𝐁 minus 𝐀. We do this by multiplying vector 𝐁 by the scalar two and then subtracting vector 𝐀. When multiplying a vector by a scalar, we multiply each of the components by the scalar. Therefore, two 𝐁 is equal to two, negative two, two.

We need to subtract the vector two, zero, negative two. We do this by subtracting each component individually. Two minus two is equal to zero. Negative two minus zero is equal to negative two. Finally, two minus negative two is equal to four, as subtracting negative two is the same as adding two.

We need to find the unit vector of this. We know that the unit vector 𝐕 hat is equal to one over the magnitude of 𝐕 multiplied by vector 𝐕. The magnitude of two 𝐁 minus 𝐀 is equal to the square root of zero squared plus negative two squared plus four squared. This is equal to the square root of 20, which simplifies to two root five. The unit vector is therefore equal to one over two root five multiplied by zero, negative two, four. We then multiply each individual component by one over two root five. Multiplying this by zero gives us zero. Negative two multiplied by one over two root five is equal to negative one over root five.

Rationalizing the denominator by multiplying the numerator and denominator by root five gives us negative root five over five. Finally multiplying four by one over two root five gives us two root five over five. The unit vector in the direction two 𝐁 minus 𝐀 has component zero, negative root five over five, and two root five over five.

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