Mixing
Show that the lines containing $n(s)$ and passing through $alpha(s)$ meet the $z$ axis under a constant angle...
$begingroup$
$$alpha(s)=left(a cosfrac{s}{c}, a sinfrac{s}{c}, bfrac{s}{c}right), quad s in mathbb R$$
$$n(s)=left(cosfrac{s}{c}, sinfrac{s}{c}, 0right)$$
Show that the lines containing $n(s)$ and passing through $alpha(s)$ meet the $z$ axis under a constant angle equal to $pi/2$.
The line containing $n(s)$ and passing through $alpha(s)$ is
$$alpha(s)+tn(s)$$
but I am not getting that this is orthogonal to the $z$ axis represented by the vector $(0,0,1)$.
Any help?
calculus differential-geometry
asked Jan 28 at 19:19
Al JebrAl Jebr
4,37943378
$endgroup$
|
show 5 more comments
$begingroup$
$$alpha(s)=left(a cosfrac{s}{c}, a sinfrac{s}{c}, bfrac{s}{c}right), quad s in mathbb R$$
$$n(s)=left(cosfrac{s}{c}, sinfrac{s}{c}, 0right)$$
Show that the lines containing $n(s)$ and passing through $alpha(s)$ meet the $z$ axis under a constant angle equal to $pi/2$.
The line containing $n(s)$ and passing through $alpha(s)$ is
$$alpha(s)+tn(s)$$
but I am not getting that this is orthogonal to the $z$ axis represented by the vector $(0,0,1)$.
Any help?
calculus differential-geometry
asked Jan 28 at 19:19
Al JebrAl Jebr
4,37943378
$endgroup$
$begingroup$
What does it mean to say a line is perpendicular to another line?
$endgroup$
– Ted Shifrin
Jan 28 at 19:22
$begingroup$
dot product is zero
$endgroup$
– Al Jebr
Jan 28 at 19:24
$begingroup$
Dot product of what vectors? Think carefully, and draw pictures.
$endgroup$
– Ted Shifrin
Jan 28 at 19:28
$begingroup$
dot product of any vector lying on one line and any vector lying on the other lying. Then angle between any two vectors, each lying on separate lines, should not change.
$endgroup$
– Al Jebr
Jan 28 at 19:30
$begingroup$
What does "vector lying on [a] line" mean?
$endgroup$
– Ted Shifrin
Jan 28 at 19:31
|
show 5 more comments
$begingroup$
$$alpha(s)=left(a cosfrac{s}{c}, a sinfrac{s}{c}, bfrac{s}{c}right), quad s in mathbb R$$
$$n(s)=left(cosfrac{s}{c}, sinfrac{s}{c}, 0right)$$
Show that the lines containing $n(s)$ and passing through $alpha(s)$ meet the $z$ axis under a constant angle equal to $pi/2$.
The line containing $n(s)$ and passing through $alpha(s)$ is
$$alpha(s)+tn(s)$$
but I am not getting that this is orthogonal to the $z$ axis represented by the vector $(0,0,1)$.
Any help?
calculus differential-geometry
asked Jan 28 at 19:19
Al JebrAl Jebr
4,37943378
$endgroup$
$$alpha(s)=left(a cosfrac{s}{c}, a sinfrac{s}{c}, bfrac{s}{c}right), quad s in mathbb R$$
$$n(s)=left(cosfrac{s}{c}, sinfrac{s}{c}, 0right)$$
Show that the lines containing $n(s)$ and passing through $alpha(s)$ meet the $z$ axis under a constant angle equal to $pi/2$.
The line containing $n(s)$ and passing through $alpha(s)$ is
$$alpha(s)+tn(s)$$
but I am not getting that this is orthogonal to the $z$ axis represented by the vector $(0,0,1)$.
Any help?
calculus differential-geometry
calculus differential-geometry
asked Jan 28 at 19:19
Al JebrAl Jebr
4,37943378
asked Jan 28 at 19:19
Al JebrAl Jebr
4,37943378
asked Jan 28 at 19:19
Al JebrAl Jebr
4,37943378
asked Jan 28 at 19:19
Al JebrAl Jebr
4,37943378
asked Jan 28 at 19:19
Al JebrAl Jebr
4,37943378
4,37943378
$begingroup$
What does it mean to say a line is perpendicular to another line?
$endgroup$
– Ted Shifrin
Jan 28 at 19:22
$begingroup$
dot product is zero
$endgroup$
– Al Jebr
Jan 28 at 19:24
$begingroup$
Dot product of what vectors? Think carefully, and draw pictures.
$endgroup$
– Ted Shifrin
Jan 28 at 19:28
$begingroup$
dot product of any vector lying on one line and any vector lying on the other lying. Then angle between any two vectors, each lying on separate lines, should not change.
$endgroup$
– Al Jebr
Jan 28 at 19:30
$begingroup$
What does "vector lying on [a] line" mean?
$endgroup$
– Ted Shifrin
Jan 28 at 19:31
|
show 5 more comments
$begingroup$
What does it mean to say a line is perpendicular to another line?
$endgroup$
– Ted Shifrin
Jan 28 at 19:22
$begingroup$
dot product is zero
$endgroup$
– Al Jebr
Jan 28 at 19:24
$begingroup$
Dot product of what vectors? Think carefully, and draw pictures.
$endgroup$
– Ted Shifrin
Jan 28 at 19:28
$begingroup$
dot product of any vector lying on one line and any vector lying on the other lying. Then angle between any two vectors, each lying on separate lines, should not change.
$endgroup$
– Al Jebr
Jan 28 at 19:30
$begingroup$
What does "vector lying on [a] line" mean?
$endgroup$
– Ted Shifrin
Jan 28 at 19:31
$begingroup$
What does it mean to say a line is perpendicular to another line?
$endgroup$
– Ted Shifrin
Jan 28 at 19:22
$begingroup$
What does it mean to say a line is perpendicular to another line?
$endgroup$
– Ted Shifrin
Jan 28 at 19:22
$begingroup$
dot product is zero
$endgroup$
– Al Jebr
Jan 28 at 19:24
$begingroup$
dot product is zero
$endgroup$
– Al Jebr
Jan 28 at 19:24
$begingroup$
Dot product of what vectors? Think carefully, and draw pictures.
$endgroup$
– Ted Shifrin
Jan 28 at 19:28
$begingroup$
Dot product of what vectors? Think carefully, and draw pictures.
$endgroup$
– Ted Shifrin
Jan 28 at 19:28
$begingroup$
dot product of any vector lying on one line and any vector lying on the other lying. Then angle between any two vectors, each lying on separate lines, should not change.
$endgroup$
– Al Jebr
Jan 28 at 19:30
$begingroup$
dot product of any vector lying on one line and any vector lying on the other lying. Then angle between any two vectors, each lying on separate lines, should not change.
$endgroup$
– Al Jebr
Jan 28 at 19:30
$begingroup$
What does "vector lying on [a] line" mean?
$endgroup$
– Ted Shifrin
Jan 28 at 19:31
$begingroup$
What does "vector lying on [a] line" mean?
$endgroup$
– Ted Shifrin
Jan 28 at 19:31
|
show 5 more comments
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$begingroup$
What does it mean to say a line is perpendicular to another line?
$endgroup$
– Ted Shifrin
Jan 28 at 19:22
$begingroup$
dot product is zero
$endgroup$
– Al Jebr
Jan 28 at 19:24
$begingroup$
Dot product of what vectors? Think carefully, and draw pictures.
$endgroup$
– Ted Shifrin
Jan 28 at 19:28
$begingroup$
dot product of any vector lying on one line and any vector lying on the other lying. Then angle between any two vectors, each lying on separate lines, should not change.
$endgroup$
– Al Jebr
Jan 28 at 19:30
$begingroup$
What does "vector lying on [a] line" mean?
$endgroup$
– Ted Shifrin
Jan 28 at 19:31