Calculus Calculus Math Forum

April 5th, 2018, 10:07 AM   #1
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Joined: Apr 2018
From: Wales

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Hello all,
I have recently been advancing my knowledge of mathematics by working through worksheets online. However, I am stumped at these particular questions, and have no clue where to begin and answer! Any chance of any answers? Answers would be appreciated as I can work through the steps logically to see how it works. All responses are highly appreciated, Thank you everyone! P.S I have added a file attachment of the questions below which is a little clearer than what I have typed out.

The dot scalar product (M) of two directional paths 'x' and 'y' is mathematically defined as follows:

M = xy $\hspace{57px}$ (1)

and
xy = |x||y|cosθ (2)

where |x| is the magnitude of directional path 'x' and |y| is the magnitude of directional path 'y' and θ is the angle between paths 'x' and 'y'

Generally, for two directional paths 'a' and 'b' defined as follows:

a = a$_1$i + a$_2$j (3)
b = b$_1$i + b$_2$j (4)

The following formulas are given for the dot or scalar product of ‘a’ and ‘b’ and their respective magnitudes. Remember the notations ‘i’ and ‘j’ represent the spatial direction of the paths.

ab = (a$_1$b$_1$) + (a$_2$b$_2$) (5)

|a| = √(a$_1^2$ + a$_2^2$) (6)

|b| = √(b$_1^2$ + b$_2^2$) (7)

If the directional paths ‘x’ and ‘y’ are defined as follows:

x = 3i + 6j (8)
y = 8i - 2j (9)

Question a.
Solve for M by interpreting all the given formulas in equations (1) to (9).

Question b.
Solve for the angle between the directional paths ‘x’ and ‘y’ by making θ the subject of the formula in equation (2).
Attached Images
 Formula math questions.jpg (13.1 KB, 5 views)

Last edited by skipjack; April 5th, 2018 at 10:44 AM.

 April 5th, 2018, 10:46 AM #2 Global Moderator   Joined: Dec 2006 Posts: 20,934 Thanks: 2208 Your attachment is a bit fuzzy.
 April 5th, 2018, 12:37 PM #3 Global Moderator   Joined: May 2007 Posts: 6,807 Thanks: 717 To get M, use equation (5). To get $cos(\theta)$, use equations (5), (6), and (7) and insert answers into equation (2).
April 5th, 2018, 01:06 PM   #4
Math Team

Joined: Jan 2015
From: Alabama

Posts: 3,264
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Quote:
 Originally Posted by jordthebrave Hello all, I have recently been advancing my knowledge of mathematics by working through worksheets online. However, I am stumped at these particular questions, and have no clue where to begin and answer! Any chance of any answers? Answers would be appreciated as I can work through the steps logically to see how it works. All responses are highly appreciated, Thank you everyone! P.S I have added a file attachment of the questions below which is a little clearer than what I have typed out. The dot scalar product (M) of two directional paths 'x' and 'y' is mathematically defined as follows: M = x∙y $\hspace{57px}$ (1) and x∙y = |x||y|cosθ (2) where |x| is the magnitude of directional path 'x' and |y| is the magnitude of directional path 'y' and θ is the angle between paths 'x' and 'y' Generally, for two directional paths 'a' and 'b' defined as follows: a = a$_1$i + a$_2$j (3) b = b$_1$i + b$_2$j (4) The following formulas are given for the dot or scalar product of ‘a’ and ‘b’ and their respective magnitudes. Remember the notations ‘i’ and ‘j’ represent the spatial direction of the paths. a∙b = (a$_1$b$_1$) + (a$_2$b$_2$) (5) |a| = √(a$_1^2$ + a$_2^2$) (6) |b| = √(b$_1^2$ + b$_2^2$) (7) If the directional paths ‘x’ and ‘y’ are defined as follows: x = 3i + 6j ( y = 8i - 2j (9) Question a. Solve for M by interpreting all the given formulas in equations (1) to (9).
Using (1) m= (8i+ 6j).(8i-2j)= 8(+ (6)(-2)= 64- 12= 52.

Using (2), |8i+ 6j|= sqrt(8(+ 6(6))= sqrt{64+ 36}= sqrt{100}= 10, |8i- 2j|= sqrt(8(+ 2(2))= sqrt(64+ 4)= sqrt(6= 2\sqrt(17)
so (8i+ 6j).(8i+ 2j)= 20sqrt(17)cos(θ) where θ is the angle between the two vectors.

Question b.
Solve for the angle between the directional paths ‘x’ and ‘y’ by making θ the subject of the formula in equation (2).[/QUOTE]
From the two different ways of finding the dot product, 20sqrt(17)cos(θ)= 52. θ= arcos(52/(20sqrt(17)). That's about 51 degrees.

 April 5th, 2018, 01:12 PM #5 Newbie   Joined: Apr 2018 From: Wales Posts: 2 Thanks: 0 Thanks for all responses, highly appreciated!

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