What does this mean (involves vectors)? I'll pick a best answer TODAY!?

Question

Okay, it says: Find the unite vectors u = (cos X)i + (sin X)j in the same direction as:

1. -i + 3j

2. 6i - 8j

I have no idea what that means, as I have never dealt with a problem like that before. I have no idea even where to begin because I do not know what it is asking me to do!! Can someone please help me figure this out?! Thank you so much and I'll pick a best answer TODAY!!!!!!!!!!!!!!!!!!!!!!

Answer 1

One option is that vector i stands for what we mere mortals refer to as the x-axis and j stands for the y axis.

i and j are orthogonal (90 degrees and same scale).

-i is a vector of length 1 in the direction of minus i.
3 j is a vector of length 3 in the direction of plus j.

It (-i + 3j) lies in the second quadrant (up and left), with angle X between pi/2 and pi (or between 90 and 180 if you are radian-challenged).

The tangent of the angle X is -3, therefore the angle X is arctan(-3) = 1.8925 radian (= 108.435 deg.).

CosX = -0.3162278
SinX = +0.9486833

The vector (cos X)i + (sin X)j is in the same direction as the resulting vector, but it measures only 1 (while the resultant measures √10).

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We could have found it faster with:

-i measures 1
3j measures 3.
Resultant measures SQRT( 1^2 + 3¨2) = SQRT(1+9) = SQRT(10) = √10

Unit vector along the resultant:
(-i + 3j) / √10 = -0.3162278i + 0.9486833j

Answer 2

I'll do a similar problem, and you can try the two you have.

suppose we were given 3i + 5j

where i is the unit vector along the x axis j is the unit vector along the y axis.

So, draw x and y axes, and plot the vector from ( 0, 0 ) to (3 , 5)

Label the sides of the right triangle ( base = 3, height = 5) and find the hypotenuse (length of the vector 3i + 5j), which is sqrt(34).

the unit vector in question is just the given vector divided by its length: (3/sqrt(34)) i + (5/sqrt(34))j

And those two coefficients can be interpreted as the cosine and sine of the angle X which the vector makes with the positive x axis.

In the other quadrants, the coordinates can be negative, but so are sines and cosines of the appropriate angles outside of the first quadrant.

Answer 3

a vector ai+bj is a unit vector if a²+b²=1
a vector multiplied by a scalar s has the same direction so
s(-i + 3j) has the same direction a is a unit vector if
(-s)²+(3s)²=1 => 10s²=1 => s = 1/sqrt(10)
so the answer to question 1 is -i/sqrt(10) + 3j/sqrt(10)
for question two :
(6s)²+(-8s)²=1 => 100s²=1 => s=1/10
SO 0,6i-0,8j is a unit vector in the same direction

Answer 4

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Answer 5

hi This definition as well only involves vectors and therefore does not require reference systems. 5.4.