# Why do we solve i and j components of a vector using trig?

• ALRedEye
In summary, when you walk 3 miles north and 4 miles west, you can get back to where you started by walking 5 miles diagonally to the southeast.
ALRedEye

## Homework Statement

I'm having trouble understanding why we solve vector components (i and j, or the horizontal and vertical legs) like a right triangle?
An example would be a 5-4-3 triangle. If 5 N was the force vector I am solving for then I would end up with 4 N in the horizontal direction and 3 N in the vertical direction. The part I don't understand is how the sum of the legs can equal more than the original force? To me it seems like i^2 + j^2 > (the original force vector)^2 and i + j = (the original force vector) makes more sense.
When I try and relate this to the real world I'm thinking maybe that 5 N force is me pushing on a box at a -53 degree angle. So that means the box is moving at 4 N along the floor and 3 N into the floor, but I'm misunderstanding how these components relate to each other since the total of the two components is 7 N and I'm only pushing at 5 N.
I'd really appreciate some help wrapping my head around this!
(Sorry if I posted to the wrong category)

ALRedEye said:

## Homework Statement

I'm having trouble understanding why we solve vector components (i and j, or the horizontal and vertical legs) like a right triangle?
An example would be a 5-4-3 triangle. If 5 N was the force vector I am solving for then I would end up with 4 N in the horizontal direction and 3 N in the vertical direction. The part I don't understand is how the sum of the legs can equal more than the original force?
What, exactly do you mean by "the sum of the legs"? The sum of the horizontal and vertical force vectors are equal to the total force- that's the whole point.

To me it seems like i^2 + j^2 > (the original force vector)^2
Assuming you mean the dot product here, (3i).(3i)= 9 and (4i).(4i)= 16. 9+ 16= 25= 5^2. Where did you get the idea that it was larger than
(the original force vector)^2

and i + j = (the original force vector) makes more sense.
Did you not mean to write 3i+ 4j= (the original force vector)?

When I try and relate this to the real world I'm thinking maybe that 5 N force is me pushing on a box at a -53 degree angle. So that means the box is moving at 4 N along the floor and 3 N into the floor
You mean there would be a 4N force pushing the box and a 3N force pressing it into the floor, right?

, but I'm misunderstanding how these components relate to each other since the total of the two components is 7 N and I'm only pushing at 5 N.
I'd really appreciate some help wrapping my head around this!
Your mistake is thinking that "the total of the two components is 7 N". You do not add the magnitudes of two vectors- you add the vectors themselves. These two vectors, of magnitude 3 and 4, add to a vector of magnitude 5. That is because the magnitude of a vector of the form ai+ bj is $\sqrt{a^2+ b^2}$.

(Sorry if I posted to the wrong category)

## The Attempt at a Solution

Do you understand that if you walk 3 miles north and 4 miles west that you can get back where you started by walking 5 miles diagonally to the southeast?

Chestermiller and Nathanael
That makes more sense. Thanks

## 1. Why do we use trigonometry to solve i and j components of a vector?

Trigonometry is used to solve the i and j components of a vector because vectors have both magnitude and direction. Trigonometry is a branch of mathematics that deals with relationships between the sides and angles of triangles, making it a useful tool for calculating and understanding direction and magnitude in vectors.

## 2. Can't we solve vectors using basic algebra instead of trigonometry?

While basic algebra can be used to solve some vector problems, trigonometry is essential for solving more complex vector problems. Trigonometry provides a more intuitive and accurate way to determine direction and magnitude, especially when dealing with angles and non-linear motion.

## 3. How do we use trigonometry to solve i and j components of a vector?

To solve for the i and j components of a vector using trigonometry, we use the trigonometric ratios of sine, cosine, and tangent. These ratios help us determine the relationship between the sides and angles of a triangle, which can then be applied to determine the direction and magnitude of the vector.

## 4. Why do we need to break down a vector into its i and j components?

Breaking down a vector into its i and j components allows us to analyze and understand the vector in terms of its horizontal and vertical components. This makes it easier to solve for the vector's direction and magnitude, as well as to add or subtract vectors using vector addition and subtraction.

## 5. Are there any real-world applications of solving i and j components of a vector using trigonometry?

Yes, there are many real-world applications of solving i and j components of a vector using trigonometry. For example, it is used in navigation and engineering to determine the direction and magnitude of forces and velocities, as well as in physics and mechanics to analyze the motion of objects in two-dimensional space.

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