angles between lines - C4
Hello people, again! 
I was originally confused by the idea of an angle in |R3, so I looked at Khanacademy's linear algebra playlist, and found https://www.khanacademy.org/math/linear-algebra/vectors_and_spaces/dot_cross_products/v/defining-the-angle-between-vectors. It was made clear to me that the angle between any two vectors is the same as the angle in a triangle with lengths corresponding to the vectors' magnitudes - |A|, |B|, |A-B|.The cosine rule could then be used to find the angle, since we'd know the lengths of the all of rays.
I cannot understand how this definition of an angle is applied to lines defined by a position vector + a multiple of a direction vector - a line is not a single point!
I am sorry for asking so many questions, but I am teaching myself C4 a year early; TheStudentRoom is my best resource for information ATM.
I was originally confused by the idea of an angle in |R3, so I looked at Khanacademy's linear algebra playlist, and found https://www.khanacademy.org/math/linear-algebra/vectors_and_spaces/dot_cross_products/v/defining-the-angle-between-vectors. It was made clear to me that the angle between any two vectors is the same as the angle in a triangle with lengths corresponding to the vectors' magnitudes - |A|, |B|, |A-B|.The cosine rule could then be used to find the angle, since we'd know the lengths of the all of rays.
I cannot understand how this definition of an angle is applied to lines defined by a position vector + a multiple of a direction vector - a line is not a single point!
I am sorry for asking so many questions, but I am teaching myself C4 a year early; TheStudentRoom is my best resource for information ATM.
(edited 10 years ago)
Original post by dire wolf
Hello people, again!
I was originally confused by the idea of an angle in |R3, so I looked at Khanacademy's linear algebra playlist, and found https://www.khanacademy.org/math/linear-algebra/vectors_and_spaces/dot_cross_products/v/defining-the-angle-between-vectors. It was made clear to me that the angle between any two vectors is the same as the angle in a triangle with lengths corresponding to the vectors' magnitudes - |A|, |B|, |A-B|.The cosine rule could then be used to find the angle, since we'd know the lengths of the all of rays.
I cannot understand how this definition of an angle is applied to lines defined by a position vector + a multiple of a direction vector - a line is not a single point!
I am sorry for asking so many questions, but I am teaching myself C4 a year early; TheStudentRoom is my best resource for information ATM.
I was originally confused by the idea of an angle in |R3, so I looked at Khanacademy's linear algebra playlist, and found https://www.khanacademy.org/math/linear-algebra/vectors_and_spaces/dot_cross_products/v/defining-the-angle-between-vectors. It was made clear to me that the angle between any two vectors is the same as the angle in a triangle with lengths corresponding to the vectors' magnitudes - |A|, |B|, |A-B|.The cosine rule could then be used to find the angle, since we'd know the lengths of the all of rays.
I cannot understand how this definition of an angle is applied to lines defined by a position vector + a multiple of a direction vector - a line is not a single point!
I am sorry for asking so many questions, but I am teaching myself C4 a year early; TheStudentRoom is my best resource for information ATM.
You just use the scalar product of the two direction vectors,
Original post by brianeverit
You just use the scalar product of the two direction vectors,
THat result does not make sense to me
Original post by dire wolf
Hello people, again!
I was originally confused by the idea of an angle in |R3, so I looked at Khanacademy's linear algebra playlist, and found https://www.khanacademy.org/math/linear-algebra/vectors_and_spaces/dot_cross_products/v/defining-the-angle-between-vectors. It was made clear to me that the angle between any two vectors is the same as the angle in a triangle with lengths corresponding to the vectors' magnitudes - |A|, |B|, |A-B|.The cosine rule could then be used to find the angle, since we'd know the lengths of the all of rays.
I cannot understand how this definition of an angle is applied to lines defined by a position vector + a multiple of a direction vector - a line is not a single point!
I am sorry for asking so many questions, but I am teaching myself C4 a year early; TheStudentRoom is my best resource for information ATM.
I was originally confused by the idea of an angle in |R3, so I looked at Khanacademy's linear algebra playlist, and found https://www.khanacademy.org/math/linear-algebra/vectors_and_spaces/dot_cross_products/v/defining-the-angle-between-vectors. It was made clear to me that the angle between any two vectors is the same as the angle in a triangle with lengths corresponding to the vectors' magnitudes - |A|, |B|, |A-B|.The cosine rule could then be used to find the angle, since we'd know the lengths of the all of rays.
I cannot understand how this definition of an angle is applied to lines defined by a position vector + a multiple of a direction vector - a line is not a single point!
I am sorry for asking so many questions, but I am teaching myself C4 a year early; TheStudentRoom is my best resource for information ATM.
If I'm understanding your confusion correctly (the bolded part above), a vector equation doesn't describe a point on a line, it describes every point on a line. The position vector gives a point on the line, and then the product of the direction vector and a scalar defines how far along the line a point is, based on the value of the scalar.
Consider the point where the two vector lines intercept, from that point the direction the lines are going in is the only thing that matters, as it's the direction that determines the angles. The angle you want is the angle inbetween them and the dot (or scalar) product relationship is what you should be using.
I'm hoping this is helpful, otherwise I've completely mis-interpreted your problem!
Original post by dire wolf
Hello people, again!
I was originally confused by the idea of an angle in |R3, so I looked at Khanacademy's linear algebra playlist, and found https://www.khanacademy.org/math/linear-algebra/vectors_and_spaces/dot_cross_products/v/defining-the-angle-between-vectors. It was made clear to me that the angle between any two vectors is the same as the angle in a triangle with lengths corresponding to the vectors' magnitudes - |A|, |B|, |A-B|.The cosine rule could then be used to find the angle, since we'd know the lengths of the all of rays.
I cannot understand how this definition of an angle is applied to lines defined by a position vector + a multiple of a direction vector - a line is not a single point!
I am sorry for asking so many questions, but I am teaching myself C4 a year early; TheStudentRoom is my best resource for information ATM.
I was originally confused by the idea of an angle in |R3, so I looked at Khanacademy's linear algebra playlist, and found https://www.khanacademy.org/math/linear-algebra/vectors_and_spaces/dot_cross_products/v/defining-the-angle-between-vectors. It was made clear to me that the angle between any two vectors is the same as the angle in a triangle with lengths corresponding to the vectors' magnitudes - |A|, |B|, |A-B|.The cosine rule could then be used to find the angle, since we'd know the lengths of the all of rays.
I cannot understand how this definition of an angle is applied to lines defined by a position vector + a multiple of a direction vector - a line is not a single point!
I am sorry for asking so many questions, but I am teaching myself C4 a year early; TheStudentRoom is my best resource for information ATM.
First, you know that the angle will remain the same if the triangle is enlarged
Second, you understand the the equation of the line gives the vector from (0,0,0) to any point on the line
Thirdly, you realise that the direction vector gives the vector AB where A and B are 2 points on the line
So, you have a point, A, where the lines meet
The direction vector from one of the lines is AB
The direction vector from the other line is AC
And there is your triangle
Original post by dire wolf
THat result does not make sense to me
Why doesn't it? It is the standard method.
Firstly, thank you all.
This post was my favourite, though. You have been very helpful to me and I will give you a rep in return. 💯
Original post by TenOfThem
First, you know that the angle will remain the same if the triangle is enlarged
Second, you understand the the equation of the line gives the vector from (0,0,0) to any point on the line
Thirdly, you realise that the direction vector gives the vector AB where A and B are 2 points on the line
So, you have a point, A, where the lines meet
The direction vector from one of the lines is AB
The direction vector from the other line is AC
And there is your triangle
Second, you understand the the equation of the line gives the vector from (0,0,0) to any point on the line
Thirdly, you realise that the direction vector gives the vector AB where A and B are 2 points on the line
So, you have a point, A, where the lines meet
The direction vector from one of the lines is AB
The direction vector from the other line is AC
And there is your triangle
This post was my favourite, though. You have been very helpful to me and I will give you a rep in return. 💯
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