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

# Question Video: Finding the Unknown Component of a Vector Parallel to Another Vector Mathematics • First Year of Secondary School

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Given that ๐ = โ๐ข โ 2๐ฃ and ๐ = ๐๐ข โ 8๐ฃ and ๐ โฅ ๐, where ๐ข and ๐ฃ are two perpendicular unit vectors, find the value of ๐.

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### Video Transcript

Given that the vector ๐ is equal to negative ๐ข minus two ๐ฃ and the vector ๐ is equal to ๐๐ข minus eight ๐ฃ and the vector ๐ is parallel to the vector ๐, where ๐ข and ๐ฃ are two perpendicular unit vectors, find the value of the scalar ๐.

In this question, weโre given some information about two vectors, the vector ๐ 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 is parallel to the vector ๐. We need to use this to determine the value of the scalar ๐.

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 is parallel to the vector ๐, there must exist some scalar ๐ such that ๐ 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 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๐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๐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. 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 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 by equating the components of both vectors in the ๐ข-direction.

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

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