Question Video: Finding the Magnitude of Two Vectors and Their Sum

Consider the vectors ๐ฎ = <2, 3> and ๐ฏ = <4, 6>. What is the magnitude of ๐ฎ? Give your answer to 2 d.p. What is the magnitude of ๐ฏ? Give your answer to 2 d.p. What is the magnitude of ๐ฎ + ๐ฏ? Give your answer to 2 d.p.

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Video Transcript

Consider the vectors ๐ฎ equals two, three and ๐ฏ equals four, six. What is the magnitude of vector ๐ฎ? What is the magnitude of vector ๐ฏ? What is the magnitude of vector ๐ฎ plus vector ๐ฏ? In all three questions we need to give our answer to two decimal places where appropriate.

We recall that the magnitude of any vector ๐ฐ can be found by square rooting ๐‘Ž squared plus ๐‘ squared, where ๐‘Ž and ๐‘ are the two components of the vector. In vector ๐ฎ, ๐‘Ž is equal to two and ๐‘ is equal to three. Whereas in vector ๐ฏ, ๐‘Ž is equal to four and ๐‘ is equal to six. The magnitude of vector ๐ฎ is therefore equal to the square root of two squared plus three squared. As two squared is equal to four and three squared is equal to nine, the magnitude of vector ๐ฎ is equal to the square root of 13. We would often leave this in surd or radical form. However, in this case, weโ€™re asked to give our answer to two decimal places. The square root of 13 is equal to 3.605551 and so on.

To round to two decimal places, our key or deciding number will be the first five. This will round our answer up. The magnitude of vector ๐ฎ to two decimal places is 3.61. We can repeat this process to calculate the magnitude of vector ๐ฏ. Four squared is equal to 16, and six squared is equal to 36. Therefore, the magnitude of vector ๐ฏ is the square root of 52. Typing this into the calculator gives us 7.211102 and so on. This time our deciding number is a one. As this is less than five, weโ€™ll round down. The magnitude of vector ๐ฏ is therefore equal to 7.21.

The final part of our question asks us to work out the magnitude of ๐ฎ plus ๐ฏ. Our first step here will be to calculate the vector ๐ฎ plus ๐ฏ. We do this by adding the corresponding components. Two plus four is equal to six, and three plus six is equal to nine. We can then calculate the magnitude of ๐ฎ plus ๐ฏ in the same way. This is equal to the square root of six squared plus nine squared. Six squared is equal to 36, and nine squared is equal to 81. Therefore, the magnitude of ๐ฎ plus ๐ฏ is equal to the square root of 117. Typing this into the calculator gives us 10.816653. The deciding number here is a six, and anything five or greater means that we round up. The magnitude of ๐ฎ plus ๐ฏ is equal to 10.82.

When looking at our three answers, you might think youโ€™ve spotted a pattern, as 3.61 plus 7.21 is equal to 10.82. This suggests that the magnitude of ๐ฎ plus ๐ฏ is equal to the magnitude of ๐ฎ plus the magnitude of ๐ฏ. This, however, is not normally the case. The only reason this works in this question is that vector ๐ฏ is actually a multiple of vector ๐ฎ. Two multiplied by two is equal to four. And three multiplied by two is equal to six.

Therefore, vector ๐ฏ is actually two lots or two multiplied by vector ๐ฎ. This in turn means that the magnitude of vector ๐ฏ is twice the magnitude of vector ๐ฎ. Root 52 is equal to two root 13. The magnitude of ๐ฎ plus ๐ฏ in this question becomes three times the magnitude of vector ๐ฎ. It is important to note, however, as previously mentioned, this will not hold for the majority of vector questions.

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