Question Video: Finding an Unknown Coefficient from Two Parallel Vectors Mathematics

If ๐šจ= ๐ข + 5๐ฃ and ๐šฉ = โˆ’20๐ข + ๐ฟ๐ฃ are two parallel vectors, then ๐ฟ = ๏ผฟ. [A] 10 [B] โˆ’10 [C] 100 [D] โˆ’100

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Video Transcript

If the vector ๐šจ is equal to ๐ข plus five ๐ฃ and the vector ๐šฉ is equal to negative 20๐ข plus ๐ฟ๐ฃ are two parallel vectors, then ๐ฟ is equal to blank. Is it option (A) 10, option (B) negative 10, option (C) 100, or is it option (D) negative 100?

In this question, weโ€™re given two vectors in terms of the unit directional vectors. Weโ€™re given the vector ๐šจ and the vector ๐šฉ. Weโ€™re told that these are two parallel vectors. We need to use this to determine the unknown value of the scalar ๐ฟ. To answer this question, letโ€™s start by recalling what it means for two vectors to be parallel. We say that the vectors ๐ฎ and ๐ฏ are parallel if ๐ฎ is a scalar multiple of vector ๐ฏ. In other words, there needs to exist some scalar ๐‘˜ such that ๐ฎ is equal to ๐‘˜ times ๐ฏ.

Since weโ€™re told the vectors ๐šจ and ๐šฉ are parallel, we can apply this definition to the two vectors. We know that there must exist some scalar value of ๐‘˜ such that vector ๐šจ is equal to ๐‘˜ times vector ๐šฉ. We can then substitute in our expressions for vectors ๐šจ and ๐šฉ in terms of the unit directional vectors. We get that ๐ข plus five ๐ฃ will be equal to ๐‘˜ times negative 20๐ข plus ๐ฟ times ๐ฃ. We can simplify the right-hand side of this expression by recalling that scalar multiplication of vectors is done component-wise. We just need to multiply each of the coefficients of ๐ข and ๐ฃ by our scalar ๐‘˜. Doing this, we get that ๐ข plus five ๐ฃ will be equal to negative 20๐‘˜๐ข plus ๐ฟ๐‘˜๐ฃ.

Remember, we know that the left-hand side and the right-hand side of this equation are equal because the vectors are parallel. And for two vectors to be equal, their components must be equal. For example, the coefficients of ๐ข on both sides of the equation must be equal. On the left-hand side of our equation, the coefficient of ๐ข is equal to one. On the right-hand side of the equation, the coefficient of ๐ข is negative 20๐‘˜. Therefore, by equating the coefficients of ๐ข, we get that one must be equal to negative 20๐‘˜. We can rearrange this for ๐‘˜ to find that ๐‘˜ must be equal to negative one divided by 20. We can then substitute this value of ๐‘˜ back into our equation. This gives us that ๐ข plus five ๐ฃ will be equal to negative 20 multiplied by negative one divided by 20๐ข plus ๐ฟ multiplied by negative one twentieth ๐ฃ.

We can, of course, simplify the right-hand side of this equation. Negative 20 multiplied by negative one divided by 20 is equal to one. However, this is not necessary. All we need to do is find the value of ๐ฟ. And we can do this by equating the coefficients of ๐ฃ on both sides of our equation. Doing this and simplifying, we get that five must be equal to negative ๐ฟ divided by 20. We can then find the value of ๐ฟ by multiplying both sides of the equation by negative 20. This gives us that ๐ฟ will be equal to five multiplied by negative 20, which is equal to negative 100.

Therefore, we were able to show if ๐šจ is the vector ๐ข plus five ๐ฃ and ๐šฉ is the vector negative 20๐ข plus ๐ฟ๐ฃ are two parallel vectors, then the value of ๐ฟ must be equal to negative 100, which was option (D).

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