#### 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 = xâˆ™y $\hspace{57px}$ (1)

and
xâˆ™y = |x||y|cosÎ¸ (2)

where |x| is the magnitude of directional path 'x' and |y| is the magnitude of directional path 'y' and Î¸ 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 â€˜aâ€™ and â€˜bâ€™ and their respective magnitudes. Remember the notations â€˜iâ€™ and â€˜jâ€™ represent the spatial direction of the paths.

aâˆ™b = (a$_1$b$_1$) + (a$_2$b$_2$) (5)

|a| = âˆš(a$_1^2$ + a$_2^2$) (6)

|b| = âˆš(b$_1^2$ + b$_2^2$) (7)

If the directional paths â€˜xâ€™ and â€˜yâ€™ 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 â€˜xâ€™ and â€˜yâ€™ by making Î¸ the subject of the formula in equation (2).

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#### skipjack

Forum Staff
Your attachment is a bit fuzzy.

#### mathman

Forum Staff
To get M, use equation (5). To get $cos(\theta)$, use equations (5), (6), and (7) and insert answers into equation (2).

#### Country Boy

Math Team
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 = xâˆ™y $\hspace{57px}$ (1)

and
xâˆ™y = |x||y|cosÎ¸ (2)

where |x| is the magnitude of directional path 'x' and |y| is the magnitude of directional path 'y' and Î¸ 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 â€˜aâ€™ and â€˜bâ€™ and their respective magnitudes. Remember the notations â€˜iâ€™ and â€˜jâ€™ represent the spatial direction of the paths.

aâˆ™b = (a$_1$b$_1$) + (a$_2$b$_2$) (5)

|a| = âˆš(a$_1^2$ + a$_2^2$) (6)

|b| = âˆš(b$_1^2$ + b$_2^2$) (7)

If the directional paths â€˜xâ€™ and â€˜yâ€™ 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).
Using (1) m= (8i+ 6j).(8i-2j)= 8(8)+ (6)(-2)= 64- 12= 52.

Using (2), |8i+ 6j|= sqrt(8(8)+ 6(6))= sqrt{64+ 36}= sqrt{100}= 10, |8i- 2j|= sqrt(8(8)+ 2(2))= sqrt(64+ 4)= sqrt(68)= 2\sqrt(17)
so (8i+ 6j).(8i+ 2j)= 20sqrt(17)cos(Î¸) where Î¸ is the angle between the two vectors.

Question b.
Solve for the angle between the directional paths â€˜xâ€™ and â€˜yâ€™ by making Î¸ the subject of the formula in equation (2).[/QUOTE]
From the two different ways of finding the dot product, 20sqrt(17)cos(Î¸)= 52. Î¸= arcos(52/(20sqrt(17)). That's about 51 degrees.

#### jordthebrave

Thanks for all responses, highly appreciated!