vector (1 Viewer)

[juantheron]

 

[Carrotsticks]

The expression |(u x v) . w| = 1/2 is the Scalar Triple Product, which computes the volume of the Parallelpipid spanned by the 3 vectors u, v and w, where in this case u and v lie on the same plane (since its u x v). So we are given that the volume of the Parallelpipid is 1/2 (so it is not the degenerate case where the Parallelpipid has volume 0)

If w x u = v - w (ie: it lies on the same plane as v and w), then this implies that the vector u is perpendicular to the vector v.

If this is the case, then it is trivial that the Dot Product is equal to 0.

Here is a diagram to demonstrate this:

Also, the HSC Syllabus does not include Dot or Cross Products of vectors. Next time if you have a question involving topics out of the HSC Syllabus, feel free to post them up in this section: http://community.boredofstudies.org/forumdisplay.php?f=1003

 

[juantheron]

Thanks moderator but I did not Understand the line

w x u = v - w (ie: it lies on the same plane as v and w )

Thanks

 

[Carrotsticks]

The Cross Product between two vectors will yield a vector that is orthogonal to the BOTH of them. So for example if I get a vector along the X axis and CROSS PRODUCT it with a vector along the Y axis, then I will get a vector along the positive/negative Z axis (depending on whether it's X x Y or Y x X).

So suppose I have 3 vectors a, b and c such that a x b = c.

I can safely say that a.c = 0 and b.c = 0 as well since a and b are both orthogonal to c.

In the diagram below, the red line is the result of the cross product (could point the other way around but either way, orthogonality is preserved).

So we have w x u = v - w (I have shown it in red in the diagram below):

Let the intersections of the 3 black lines (so where I drew the right angle symbol) be O.

By definition of the cross product, we have the plane containing WOU being perpendicular to the plane containing WOV. But vector V lies on plane WOV and vector U lies on WOU, so the two vectors are perpendicular.

So U is perpendicular to V, so therefore U . V = 0.

Really, I should have drawn the parallelpipid to be a rectangular prism (which is a special case of a parallelpipid where all 3 vectors are orthogonal to each other such that they make 'their own XYZ plane' if you get what I mean).

 

[juantheron]

Thanks moderator got it