Video Transcript
If the vector ๐จ is equal to ๐ข
plus five ๐ฃ and the vector ๐ฉ is equal to negative 20๐ข plus ๐ฟ๐ฃ are two parallel
vectors, then ๐ฟ 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, weโre given two
vectors in terms of the unit directional vectors. Weโre given the vector ๐จ and the
vector ๐ฉ. Weโre told that these are two
parallel vectors. We need to use this to determine
the unknown value of the scalar ๐ฟ. To answer this question, letโs
start by recalling what it means for two vectors to be parallel. We say that the vectors ๐ฎ and ๐ฏ
are parallel if ๐ฎ is a scalar multiple of vector ๐ฏ. In other words, there needs to
exist some scalar ๐ such that ๐ฎ is equal to ๐ times ๐ฏ.
Since weโre told the vectors ๐จ and
๐ฉ 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 ๐จ is equal to ๐ times vector ๐ฉ. We can then substitute in our
expressions for vectors ๐จ and ๐ฉ in terms of the unit directional vectors. We get that ๐ข plus five ๐ฃ will be
equal to ๐ times negative 20๐ข plus ๐ฟ times ๐ฃ. 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 ๐ข and ๐ฃ by our scalar ๐. Doing this, we get that ๐ข plus
five ๐ฃ will be equal to negative 20๐๐ข plus ๐ฟ๐๐ฃ.
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 ๐ข
on both sides of the equation must be equal. On the left-hand side of our
equation, the coefficient of ๐ข is equal to one. On the right-hand side of the
equation, the coefficient of ๐ข is negative 20๐. Therefore, by equating the
coefficients of ๐ข, 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 ๐ข plus five ๐ฃ
will be equal to negative 20 multiplied by negative one divided by 20๐ข plus ๐ฟ
multiplied by negative one twentieth ๐ฃ.
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 ๐ฟ. And we can do this by equating the
coefficients of ๐ฃ on both sides of our equation. Doing this and simplifying, we get
that five must be equal to negative ๐ฟ divided by 20. We can then find the value of ๐ฟ by
multiplying both sides of the equation by negative 20. This gives us that ๐ฟ will be equal
to five multiplied by negative 20, which is equal to negative 100.
Therefore, we were able to show if
๐จ is the vector ๐ข plus five ๐ฃ and ๐ฉ is the vector negative 20๐ข plus ๐ฟ๐ฃ are
two parallel vectors, then the value of ๐ฟ must be equal to negative 100, which was
option (D).
