Video: Finding the Magnitude of Vectors

Given that 𝐀 = <1, −2, 2>, 𝐁 = <2, −4, 𝑛> and 𝐀 and 𝐁 are two parallel vectors, find |𝐁|.

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Video Transcript

Given that 𝐀 is equal to one, negative two, two; 𝐁 is equal to two, negative four, 𝑛; and 𝐀 and 𝐁 are two parallel vectors, find the magnitude of vector 𝐁.

We begin by recalling that for two parallel vectors 𝐮 and 𝐯, 𝐯 is equal to 𝑘 multiplied by 𝐮, where 𝑘 is a scalar quantity. In this question, vector 𝐁 is therefore equal to 𝑘 multiplied by vector 𝐀. The vector two, negative four, 𝑛 is equal to 𝑘 multiplied by one, negative two, two. We can multiply a vector by a scalar by multiplying each of the components by that scalar. The right-hand side therefore becomes 𝑘, negative two 𝑘, two 𝑘.

As the two vectors are equal, each of their corresponding components must be equal. This means that two is equal to 𝑘. Negative four is equal to negative two 𝑘. Dividing both sides of this equation by negative two, we once again see that 𝑘 is equal to two. Finally, we have 𝑛 is equal to two 𝑘. As we have already worked out that 𝑘 is equal to two, 𝑛 is equal to four. This means that vector 𝐁 is equal to two, negative four, four. And we can use this to calculate the magnitude.

The magnitude of any vector is equal to the square root of the sum of the squares of its individual components. Therefore, the magnitude of vector 𝐁 is equal to the square root of two squared plus negative four squared plus four squared. Two squared is equal to four, and negative four squared as well as four squared are equal to 16. The magnitude of vector 𝐁 is therefore equal to the square root of 36. As the magnitude of any vector must always be positive, this is equal to six. If vectors 𝐀 and 𝐁 are parallel, the magnitude of vector 𝐁 is six.

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