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Find the inner product of ⃗a = (1, 45◦) and ⃗b = (2, 90◦), where these vectors are in polar coordinates (r, θ).

Thanks =) 1st post here btw.

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- Thread starter spaderdabomb
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- #1

- 49

- 0

Find the inner product of ⃗a = (1, 45◦) and ⃗b = (2, 90◦), where these vectors are in polar coordinates (r, θ).

Thanks =) 1st post here btw.

- #2

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http://www.iancgbell.clara.net/maths/vectors.htm

About midway down this page you can see that the dot product in polar coordinates is [itex]\small (r_1,\theta_1) \tiny \bullet \small (r_2,\theta_2) = r_1r_2 \cos(\theta_1-\theta_2)[/itex]. One solution is to use this formula.

The other one is just to represent the vectors in normal cartesian coordinates as [itex]\frac{1}{\sqrt{2}}(1,1)[/itex] (the factor 1/sqrt(2) is there to make the vector have length 1) and [itex](2,0)[/itex] and then use the normal dot product.

About midway down this page you can see that the dot product in polar coordinates is [itex]\small (r_1,\theta_1) \tiny \bullet \small (r_2,\theta_2) = r_1r_2 \cos(\theta_1-\theta_2)[/itex]. One solution is to use this formula.

The other one is just to represent the vectors in normal cartesian coordinates as [itex]\frac{1}{\sqrt{2}}(1,1)[/itex] (the factor 1/sqrt(2) is there to make the vector have length 1) and [itex](2,0)[/itex] and then use the normal dot product.

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- #3

HallsofIvy

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This is more calculus than "linear algebra". There are two ways to go:

1) The "non-coordinate" definition of the dot product: [itex]u\cdot v= |u||v|cos(\theta)[/itex], where [itex]\theta[/itex] is the angle between the two vectors. Here, |u|= 1, |v|= 2 and angle between them is 45 degrees.

2) Convert to Cartesian coordinates. The vector at 45 degrees with length 1 is [itex]<\sqrt{2}/2, \sqrt{2}/2>[/itex]. The vector at 90 degrees, with length 2, is <0, 1>.

1) The "non-coordinate" definition of the dot product: [itex]u\cdot v= |u||v|cos(\theta)[/itex], where [itex]\theta[/itex] is the angle between the two vectors. Here, |u|= 1, |v|= 2 and angle between them is 45 degrees.

2) Convert to Cartesian coordinates. The vector at 45 degrees with length 1 is [itex]<\sqrt{2}/2, \sqrt{2}/2>[/itex]. The vector at 90 degrees, with length 2, is <0, 1>.

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