- #1

lonewolf219

- 186

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The solution was (initial point)+[<vector>/unit vector](speed)

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- Thread starter lonewolf219
- Start date

In summary, In an example my professor gave in class, a particle moves (in a straight line) from point A to point B (starting from t=0) and traveling at 6 m/s. The solution included a unit vector. Does that mean all vector equations describing velocity must include a unit vector? No, they don't.

- #1

lonewolf219

- 186

- 2

The solution was (initial point)+[<vector>/unit vector](speed)

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- #2

Mathoholic!

- 49

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[itex]\vec{r}[/itex]=[itex]\vec{r}[/itex]

Because when t=0, [itex]\vec{r}[/itex]

And [itex]\vec{r}[/itex]

[itex]\vec{v}[/itex]=6[itex]\hat{v}[/itex]

Being [itex]\hat{v}[/itex] a unit vector (normed vector), like [itex]\hat{r}[/itex].

It's not obligatory to represent, in a vector equation describing velocity, the unit vector [itex]\hat{v}[/itex]. You can always replace it by [itex]\frac{\vec{v}}{|\vec{v}|}[/itex], which is the same thing.

However, it's advised to use them because they facilitate geometrical description of equations.

For example, the Newton's law of gravitational force can be written like this:

[itex]\vec{F}[/itex]

Instead of the usual:

[itex]\vec{F}[/itex]

The first one may lead you to think that [itex]\vec{F}[/itex]

I hope I've cleared your doubt about unit vectors, if not other users may, or else explain yourself better.

- #3

lonewolf219

- 186

- 2

r'(t)=A+<4, 5, 6>6t

If there is an initial point, a speed and a vector, what am I missing if this is incorrect?

- #4

Mathoholic!

- 49

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lonewolf219 said:

r'(t)=A+<4, 5, 6>6t

If there is an initial point, a speed and a vector, what am I missing if this is incorrect?

If |[itex]\vec{v}[/itex]| is constant through time and [itex]\vec{v}[/itex]=[itex]\vec{const}[/itex] (straight line, as you said before), then the vector velocity is expressed as:

[itex]\vec{v}[/itex]=|[itex]\vec{v}[/itex]|[itex]\hat{v}[/itex]

Now, if there's a vector parallel to the movement (thus, to the vector velocity) and the speed is 6 m/s. Firstly, you note that the magnitude of (4,5,6) is not equal to 6, otherwise, that'd be your vector velocity. But if (4,5,6) is parallel to the velocity, then:

λ(4,5,6)=[itex]\vec{v}[/itex]

You need to determine the x,y,z coordinates of [itex]\vec{v}[/itex] using that expression and knowing its magnitude (6 m/s) and most importantly, calculating λ.

And then you'll be able to express the vector velocity as:

[itex]\vec{v}[/itex]=v

And yes, it's incorrect to express the vector velocity as:

[itex]\vec{r'(t)}[/itex]=A+(4,5,6)6t

The RHS has units of m/s and LHS has at least units of m (but never m/s!). Secondly, you cannot sum points (A) with vectors of velocity, you can only do that with vectors of position ([itex]\vec{r}[/itex]).

- #5

lonewolf219

- 186

- 2

r'(t)=[<4,5,6>/square root77](6t)? That would make the equation v(t)=speed(velocity unit vector)?

r(t)= (1,2,3)+[<4,5,6>/square root77](6t)

Hope I've got it?

- #6

Mathoholic!

- 49

- 0

lonewolf219 said:

r'(t)=[<4,5,6>/square root77](6t)? That would make the equation v(t)=speed(velocity unit vector)?

r(t)= (1,2,3)+[<4,5,6>/square root77](6t)

Hope I've got it?

The vector position is well written. As for the vector velocity, not so much. If the speed is constant through time, then the vector velocity doesn't have the parameter time (t), as it is also the 1st derivative of the position. As for the rest, it's all good.

- #7

lonewolf219

- 186

- 2

Vector calculus is a branch of mathematics that deals with the study of vector fields and their derivatives. It involves the use of vector operations such as dot product, cross product, and gradient to solve problems related to motion and forces.

Velocity is a vector quantity that describes the rate of change of an object's position. Vector calculus is used to calculate the velocity of an object by taking the derivative of its position vector with respect to time.

A scalar field is a function that assigns a single value (scalar) to every point in space, while a vector field assigns a vector to every point in space. Velocity is an example of a vector field, while temperature is an example of a scalar field.

The gradient is a vector operation that represents the rate of change of a scalar field. In vector calculus, it is used to find the direction and magnitude of the steepest ascent or descent of a scalar field at a specific point.

Vector calculus has many applications in various fields, including physics, engineering, and computer graphics. It is used to study motion and forces, electric and magnetic fields, fluid dynamics, optimization problems, and 3D modeling and animation, to name a few.

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