Parallel/Orthogonal in Calc 3.

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    Calc 3
In summary, to represent vector 'w' as the sum of a vector parallel to 'v' and a vector orthogonal to 'v', the dot product of 'w' and 'v' must equal the magnitude of 'w'.
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tarheels88
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Homework Statement


Let vector 'v'=<1,-1,5> and vector 'w'=<40,-19,-1>. Represent 'w' in the form of 'w'='g'+'h',
where 'g' is parallel to 'v', and 'h' is orthogonal to 'v'. This is related to the dot product. But I don't know where to begin




Homework Equations


'v' is a vector. 'g' is a vector. 'w' is a vector. 'h' is a vector


The Attempt at a Solution


So far I set up an equation <40,-19,-1>='g'+'h'
I doubt this is right so far. I know that g has to be a multiple of v and h has to = 0 in order for it to be orthogonal.
 
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  • #2
well what do you know about dot products? in particular consider
[tex]w \bullet v [/tex]
 

FAQ: Parallel/Orthogonal in Calc 3.

What is the difference between parallel and orthogonal vectors in Calc 3?

Parallel vectors are two or more vectors that have the same or opposite direction. This means that they either point in the same direction or in completely opposite directions. On the other hand, orthogonal vectors are two vectors that are perpendicular to each other, meaning they form a 90-degree angle. In Calc 3, we use the concept of parallel and orthogonal vectors to understand and solve problems involving 3-dimensional spaces.

How do you determine if two vectors are parallel or orthogonal?

To determine if two vectors are parallel, we can use the dot product. If the dot product of two vectors is equal to the product of their magnitudes, then the vectors are parallel. To determine if two vectors are orthogonal, we can use the cross product. If the cross product of two vectors is equal to zero, then the vectors are orthogonal.

Can a vector be both parallel and orthogonal to another vector?

No, a vector cannot be both parallel and orthogonal to another vector at the same time. If two vectors are parallel, then their dot product will be non-zero and their cross product will be zero, indicating that they are not orthogonal. Similarly, if two vectors are orthogonal, then their cross product will be non-zero and their dot product will be zero, indicating that they are not parallel.

How do parallel and orthogonal vectors relate to linear independence in Calc 3?

In Calc 3, parallel and orthogonal vectors play a crucial role in determining linear independence. Two or more vectors are considered linearly independent if none of them can be written as a linear combination of the others. In other words, if two vectors are parallel, they are not linearly independent. However, if two vectors are orthogonal, they are always linearly independent.

How are parallel and orthogonal vectors used in real-world applications?

In real-world applications, parallel and orthogonal vectors are used in various fields such as engineering, physics, and computer graphics. For example, in engineering, parallel and orthogonal vectors are used to represent forces and motion in 3-dimensional systems. In computer graphics, orthogonal vectors are used to determine the orientation of objects in 3D space, while parallel vectors are used to create smooth and realistic lighting effects.

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