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topjm [15]
3 years ago
8

Find two unit vectors orthogonal to both 8, 5, 1 and −1, 1, 0 .

Mathematics
1 answer:
Elina [12.6K]3 years ago
7 0
In order to do this, you must first find the "cross product" of these vectors. To do that, we can use several methods. To simplify this first, I suggest you compute:

‹1, -1, 1› × ‹0, 1, 1›

You are interested in vectors orthogonal to the originals, which don't change when you scale them. Using 0,-1,1 is much easier than 6s and 7s.

So what methods are there to compute this? You can review them here (or presumably in your class notes or textbook):
http://en.wikipedia.org/wiki/Cross_produ...

In addition to these methods, sometimes I like to set up:
‹1, -1, 1› • ‹a, b, c› = 0
‹0, 1, 1› • ‹a, b, c› = 0

That is the dot product, and having these dot products equal zero guarantees orthogonality. You can convert that to:

a - b + c = 0
b + c = 0

This is two equations, three unknowns, so you can solve it with one free parameter:

b = -c
a = c - b = -2c

The computation, regardless of method, yields:
‹1, -1, 1› × ‹0, 1, 1› = ‹-2, -1, 1›

The above method, solving equations, works because you'd just plug in c=1 to obtain this solution. However, it is not a unit vector. There will always be two unit vectors (if you find one, then its negative will be the other of course). To find the unit vector, we need to find the magnitude of our vector:

|| ‹-2, -1, 1› || = √( (-2)² + (-1)² + (1)² ) = √( 4 + 1 + 1 ) = √6

Then we divide that vector by its magnitude to yield one solution:

‹ -2/√6 , -1/√6 , 1/√6 ›

And take the negative for the other:

‹ 2/√6 , 1/√6 , -1/√6 ›
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Ray NP is angle bisector of angle MNQ and measure of angle PNQ= 2x+1. find measure of angle MNQ
statuscvo [17]

C.

An angle bisector splits angles into halves thus resulting in PNQ. The total angle is however twice the PNQ so 4x + 2.

Hope this helps.

5 0
2 years ago
A pound of turkey costs (3w 8) dollars and a pound of cheese costs (4w-3) dollars.mrs.young bought 2 pounds of turkey and three
zlopas [31]
x = 2(3w + 8) + 3(4w - 3) , where x is the total cost in terms of w.

x = (6w + 16) + (12w - 9)

x = 18w + 7
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3 years ago
Which statement could be an interpretation of the graphs x intercept or y intercept
Hatshy [7]
The x-intercept is point where the graph cuts the x-axis, while the y-intercept is the point where the graph cuts the y-axis. The y-intercept in a straight line equation normally represents some initial amount. Thus the correct answer that can be deduced from our graph is:
x-intercept is 400
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Thus the correct description is:
Ayita opened a bank account with $800
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HELP WITH THESE QUESTIONS PART 2.
polet [3.4K]

Answer: (a) IV (b) positive (c) \frac{\pi}{4} (d) C (e) \sqrt{2}

<u>Step-by-step explanation:</u>

a) \frac{15\pi}{4} - \frac{8\pi}{4} = \frac{7\pi}{4}, which is located in Quadrant IV <em>per the Unit Circle.  </em>

b) sec is \frac{1}{cos}. Cos is the x-coordinate.  The x-coordinate inQuadrant IV is positive.

c) the reference angle is the angle from \frac{7\pi}{4} to 2π = \frac{\pi}{4}

d) since the angle is below the x-axis and the reference angle is\frac{\pi}{4}, then the angle is equal to -sec(\frac{\pi}{4})

e) sec = \frac{1}{cos}   ⇒   sec \frac{7\pi}{4} = \frac{2}{\sqrt{2} } = \frac{2}{\sqrt{2} }(\frac{\sqrt{2}} {\sqrt{2}}) = \frac{2\sqrt{2} }{2} = \sqrt{2}

**********************************************************

Answer: (a) \frac{3\pi}{2} (b) (0, -1) (c) A (d) -1

<u>Step-by-step explanation:</u>

a) \frac{11\pi}{2} - \frac{4\pi}{2} = \frac{7\pi}{2} - \frac{11\pi}{2} = \frac{3\pi}{2}

b) \frac{3\pi}{2} is on the Unit Circle at (0, -1)

c) sin = \frac{opposite}{hypotenuse} which equals \frac{y}{r} on the Unit Circle.

d) sin is the y-coordinate. sin (\frac{3\pi}{2}) = -1

6 0
3 years ago
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