Please tell me how can we plot graph of position vector vs time curve

Question

If you have a position vector

$${\bf r} = x\hat{\bf i} + y\hat{\bf j} + z\hat{\bf k}$$ With respect to origin and independently each $x,y$ and $z$ coordinates are function of time then how can we depict this vector in (may be for each coordinates functions are different) position vector vs time graph

Ff we could draw a graph then how we can find velocity vector (obviously by slope of tangent), average velocity, displacement and distance covered by the particle.

Note:

1. Please do consider the fact that i am a high school student

2. i am specifically concerned with vectors so please provide me an answer which is vectorially correct.

Answer 1

You have four coordinates: time $t$, and spacial $x, y, z$. We can only graph up to 3-dimensions, and even then what we see is a 2-dimensional projection of that 3-dimensional graph. So no, we cannot do a full graph of this trajectory. There are various things we can do, such as what PM 2Ring provided, where the true 4D graph is projected onto the spacial dimensions only (and which is then projected onto a 2D plane in an arbitrary direction which you can control). Alternatively, you can plot $x, y, z$ separately against time, producing three 2D graphs. But generally this is harder for us to interpret than the 3D plot with time collapsed.

As for your other questions, these are basic formulas that you surely could find in your text book, but here they are:

• velocity vector - this is the derivative of the position vector with respect to time: $$\frac{d\mathbf r}{dt} = \frac{dx}{dt}\mathbf{\hat i} + \frac{dy}{dt}\mathbf{\hat j} + \frac{dz}{dt}\mathbf{\hat z}$$
• average velocity - this is defined over a time interval. Since you haven't specified a time interval, it is not entirely clear what average you are after. Just like any average, it is a sum of values divided by the number of values being summed. Except that the "sum" is interpreted to be the integral over an interval, and the "number of values" is interpreted to be the length of the interval: $$\text{average velocity over }[a,b] = \frac1{b-a}\int_a^b \frac{d\mathbf r}{dt}\, dt = \frac{\mathbf r(b) - \mathbf r(a)}{b-a}$$where the last equality is obtained by the fundamental theorem of calculus (assuming $\bf r$ is well-behaved, as is generally done in physics).
• displacement - displacement from what? Displacement is a relative term, and is only sensible if there is some reference to be displaced from. The displacement of the particle position at time $b$ from its position at time $a$ is just $\mathbf r(b) - \mathbf r(a)$.
• distance covered. And again, this is only defined in reference to some time interval. The distance along the curve between times $a$ and $b$ is $$\text{length } = \int_a^b \left\|\frac{d\mathbf r}{dt}\right\|\,dt = \int_a^b \sqrt{{\dot x}^2 + {\dot y}^2 + {\dot z}^2}\,dt$$ (where the dots indicate the derivatives with respect to time).