Question Video: Using the Properties of Vectors to Sum Two Vectors | Nagwa Question Video: Using the Properties of Vectors to Sum Two Vectors | Nagwa

Question Video: Using the Properties of Vectors to Sum Two Vectors Mathematics • First Year of Secondary School

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Complete the following: If ๐€ = โŒฉ1, 2โŒช and ๐ = โˆ’๐€, then ๐€ + ๐ = ๏ผฟ.

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Video Transcript

Complete the following. If ๐€ is the vector with components one, two and ๐ equals negative ๐€, then ๐€ plus ๐ is equal to what.

There are a couple of ways to answer this question. We could use scalar multiplication on our vector ๐€ equals one, two, multiplying it by negative one to get the vector ๐. And now recalling that multiplying any vector ๐ฎ by a scalar ๐‘˜ means we multiply each component of ๐ฎ by ๐‘˜, in our case, weโ€™re multiplying vector ๐€ by the scalar negative one. So here ๐‘˜ is negative one, ๐ฎ sub one is one, and ๐ฎ sub two is two. This gives the components of vector ๐ as negative one times one, which is negative one, and negative one times two, which is negative two. And so our vector ๐ has components negative one and negative two.

So now we can find the sum of vectors ๐€ and ๐ using vector addition, where for vectors ๐ฎ and ๐ฏ in two dimensions with components ๐ฎ sub one and ๐ฎ sub two and ๐ฏ sub one and ๐ฏ sub two, respectively, their sum ๐ฎ plus ๐ฏ has components ๐ฎ sub one plus ๐ฏ sub one and ๐ฎ sub two plus ๐ฏ sub two. That is, to sum the vectors, we sum each pair of components.

So now applying this to our vectors ๐€ and ๐, we have one plus negative one for the first component of the sum and two plus negative two for the second component, which is actually the zero vector with components zero, zero. So the sum of vectors ๐€ and ๐, where ๐ is negative ๐€, is the zero vector.

Now, as an alternative method to answer this question, we couldโ€™ve used just one property of vectors. Thatโ€™s the additive inverse property, which tells us that for any vector ๐ฎ, the sum of ๐ฎ with negative ๐ฎ is the zero vector. In our case, we have the sum of vectors ๐€ and ๐, which since ๐ is equal to negative ๐€ is simply ๐€ plus negative ๐€. And by the additive inverse property, this is equal to the zero vector, which as we know from before is the vector with components zero, zero.

Hence, if ๐€ is the vector with components one and two and ๐ is equal to negative ๐€, then the sum ๐€ plus ๐ is equal to the zero vector with components zero, zero.

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