Video Transcript
Given that the vector 𝐌 is equal to negative 𝐢 minus two 𝐣 and the vector 𝐋 is equal to 𝑎𝐢 minus eight 𝐣 and the vector 𝐌 is parallel to the vector 𝐋, where 𝐢 and 𝐣 are two perpendicular unit vectors, find the value of the scalar 𝑎.
In this question, we’re given some information about two vectors, the vector 𝐌 and the vector 𝐋. We’re given these vectors in terms of two perpendicular unit directional vectors, 𝐢 and 𝐣. We’re also told that the vector 𝐌 is parallel to the vector 𝐋. We need to use this to determine the value of the scalar 𝑎.
To do this, let’s start by recalling what it means for two vectors to be parallel. We say that two vectors are parallel if they’re scalar multiples of each other. In other words, we’ll say the vectors 𝐮 and 𝐯 are parallel if there exists some scalar 𝑘 such that 𝐮 is equal to 𝑘 times 𝐯. Since we’re told that the vector 𝐌 is parallel to the vector 𝐋, there must exist some scalar 𝑘 such that 𝐌 is equal to 𝑘 times 𝐋. To help us find the value of this scalar 𝑘, we can substitute the expressions we’re given for vectors 𝐌 and 𝐋 in terms of the unit directional vectors 𝐢 and 𝐣.
We get negative 𝐢 minus two 𝐣 will be equal to 𝑘 times 𝑎𝐢 minus eight 𝐣. We can then simplify the right-hand side of this equation by remembering to multiply a vector by a scalar, we multiply all of the components of our vector by our scalar, where we remember the components of our vector will be the coefficients of the unit directional vectors 𝐢 and 𝐣. This gives us that negative 𝐢 minus two 𝐣 will be equal to 𝑘𝑎𝐢 minus eight 𝑘𝐣. Remember, for two vectors to be equal, their components must be equal. So we can compare the components of 𝐣 on both sides of our equation. This then gives us the equation negative two must be equal to negative eight 𝑘, which we can solve for 𝑘 by dividing through by negative eight.
This then gives us that 𝑘 is equal to negative two divided by negative eight, which simplifies to give us one-quarter. But we’re not yet done. Remember, the question wants us to find the value of 𝑎. We can substitute our value of 𝑘 back into our equation. This then gives us the equation negative 𝐢 minus two 𝐣 will be equal to 𝑎 over four 𝐢 minus two 𝐣. Remember, for both of these vectors to be equal, their components must be equal. This means we can construct an equation for 𝑎 by equating the components of both vectors in the 𝐢-direction.
We get that negative one must be equal to 𝑎 divided by four. We can then solve this equation for 𝑎. We multiply both sides of our equation through by four, giving us that 𝑎 should be equal to negative four. Therefore, we were able to show if 𝐌 is the vector negative 𝐢 minus two 𝐣 and 𝐋 is the vector 𝑎𝐢 minus eight 𝐣 and the vector 𝐌 is parallel to the vector 𝐋, where 𝐢 and 𝐣 are two perpendicular unit vectors, then the value of 𝑎 must be equal to negative four.
