Question Video: Finding the Unknown Component of a Vector Parallel to Another Vector

Given that ๐Œ = โˆ’๐ข โˆ’ 2๐ฃ and ๐‹ = ๐‘Ž๐ข โˆ’ 8๐ฃ and ๐Œ โˆฅ ๐‹, where ๐ข and ๐ฃ are two perpendicular unit vectors, find the value of ๐‘Ž.

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Video Transcript

Given that the vector ๐Œ is equal to negative ๐ข minus two ๐ฃ and the vector ๐‹ is equal to ๐‘Ž๐ข minus eight ๐ฃ and the vector ๐Œ is parallel to the vector ๐‹, where ๐ข and ๐ฃ are two perpendicular unit vectors, find the value of the scalar ๐‘Ž.

In this question, weโ€™re given some information about two vectors, the vector ๐Œ and the vector ๐‹. Weโ€™re given these vectors in terms of two perpendicular unit directional vectors, ๐ข and ๐ฃ. Weโ€™re also told that the vector ๐Œ is parallel to the vector ๐‹. We need to use this to determine the value of the scalar ๐‘Ž.

To do this, letโ€™s start by recalling what it means for two vectors to be parallel. We say that two vectors are parallel if theyโ€™re scalar multiples of each other. In other words, weโ€™ll say the vectors ๐ฎ and ๐ฏ are parallel if there exists some scalar ๐‘˜ such that ๐ฎ is equal to ๐‘˜ times ๐ฏ. Since weโ€™re told that the vector ๐Œ is parallel to the vector ๐‹, there must exist some scalar ๐‘˜ such that ๐Œ is equal to ๐‘˜ times ๐‹. To help us find the value of this scalar ๐‘˜, we can substitute the expressions weโ€™re given for vectors ๐Œ and ๐‹ in terms of the unit directional vectors ๐ข and ๐ฃ.

We get negative ๐ข minus two ๐ฃ will be equal to ๐‘˜ times ๐‘Ž๐ข minus eight ๐ฃ. 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 ๐ข and ๐ฃ. This gives us that negative ๐ข minus two ๐ฃ will be equal to ๐‘˜๐‘Ž๐ข minus eight ๐‘˜๐ฃ. Remember, for two vectors to be equal, their components must be equal. So we can compare the components of ๐ฃ on both sides of our equation. This then gives us the equation negative two must be equal to negative eight ๐‘˜, which we can solve for ๐‘˜ by dividing through by negative eight.

This then gives us that ๐‘˜ is equal to negative two divided by negative eight, which simplifies to give us one-quarter. But weโ€™re not yet done. Remember, the question wants us to find the value of ๐‘Ž. We can substitute our value of ๐‘˜ back into our equation. This then gives us the equation negative ๐ข minus two ๐ฃ will be equal to ๐‘Ž over four ๐ข minus two ๐ฃ. Remember, for both of these vectors to be equal, their components must be equal. This means we can construct an equation for ๐‘Ž by equating the components of both vectors in the ๐ข-direction.

We get that negative one must be equal to ๐‘Ž divided by four. We can then solve this equation for ๐‘Ž. We multiply both sides of our equation through by four, giving us that ๐‘Ž should be equal to negative four. Therefore, we were able to show if ๐Œ is the vector negative ๐ข minus two ๐ฃ and ๐‹ is the vector ๐‘Ž๐ข minus eight ๐ฃ and the vector ๐Œ is parallel to the vector ๐‹, where ๐ข and ๐ฃ are two perpendicular unit vectors, then the value of ๐‘Ž must be equal to negative four.

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