Essential Specialist Mathematics Third Edition Textbook questions thread

[golden]

How would I work out questions 3, 4 and question 6 D from the chapter review (chapter 1)?

The Essential book can be found here: http://vce.atarnotes.com/forum/index.php/topic,35108.0.html.

Thanks.

[kamil9876]

What are you having trouble with?

[Milkshake]

Let a=i-2j+2k and let b be a vector, such that the vector resolute of a in the direction of b is b-hat.
a) Find the cosine of the angle between the directions of a and b
b) Find |b| if the vector resolute of b in the direction of a is 2a-hat

Help please?

[kamil9876]

a)

Draw the picture. Let be an origin and let (so ) Now let be the point such that . Because is the vector resolute of we have that the angle is a right angle. (this is easier than it sounds once you draw it)

(since b-hat is a unit vector). We can know use trig to show that .

b)

Again we will draw the picture: choose an origin and let be the point such that . Let be the point such that we see that angle is a right angle. (again, easier than it sounds once you draw it)

So now we can use trig:



but from part and

[Milkshake]

Ooh okay I get it. Thanks!

Also, what does it mean for 3 vectors to be mutually perpendicular to each other?

[golden]

What are you having trouble with?

Thanks for the reply.

With question 17, how would I prove that the lines intersect?

Please refer to below.

http://vce.atarnotes.com/forum/index.php?action=dlattach;topic=36251.0;attach=9630;image

This question is open to anyone.

[kamil9876]

I'm assuming your having trouble with b:

if we let C and D be the points such that OC=a-hat, OD=b-hat we see that we want the vector that bisects angle COD(as COD=AOB). However the triangle COD is isosceles (as |OC|=|OD|=1) hence we just want a vector that bisects the line CD. if M is the midpoint of CD then OM=OC+ 0.5CD. Figure out what this is, then "scale it" to a unit vector. (draw picture for clarity's sake).

Also, what does it mean for 3 vectors to be mutually perpendicular to each other?

Pretty sure that means "pairwise perpendicular". ie: any pair of two vectors out of the three is perpendicular, like i,j,k for example. As opposed to something like say i,j,2j; where not all pairs are perpendicular (the pair j,2j in this case) even though some are (like i,2j)

[golden]

How would I do questions c and d?

Thanks.

[kamil9876]

See a discussion here

[Milkshake]

Find the conjugate of:
-3 cis (2pi/3)
?

[xZero]

Find the conjugate of:
-3 cis (2pi/3)
?

Draw it on a complex plane and reflect it by the x-axis and you'll get the conjugate

[Milkshake]

Hmm, if I do that then i get -3cis(-2pi/3) right? But the book says the answer is 3cis(pi/3)

[xZero]

Hint: the negative sign actually changes the direction of the line

[Milkshake]

..... My god how did I not realise that.
My initial approach was expanding and then simplifying it again lol, took ages.

Thank you!

[golden]

The region, for which x ≥ 0, bounded by the curves y = cos x and y = sin x and the
y axis, is rotated around the x axis forming a solid of revolution. By using the identity
cos 2x = cos2 x − sin2 x, obtain a volume for this solid.