Question

explain co_initia/vectros

Answer 1

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Kinematics aims to provide a description of the spatial position of bodies or systems of material particles, the rate at which the particles are moving (velocity), and the rate at which their velocity is changing (acceleration). When the causative forces are disregarded, motion descriptions are possible only for particles having constrained motion—i.e., moving on determinate paths. In unconstrained, or free, motion, the forces determine the shape of the path.

For a particle moving on a straight path, a list of positions and corresponding times would constitute a suitable scheme for describing the motion of the particle. A continuous description would require a mathematical formula expressing position in terms of time.


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The amount of inertia that an object possesses is proportional to its mass . However, inertia is not the same thing as mass or momentum (the product of velocity and mass). The mass of an object can be measured by observing the extent of its inertia. This is done by measuring the amount of force required to produce a certain acceleration .

The property of inertia is useful in navigation systems. A massive object can provide a constant reference for acceleration, and in particular, changes in direction, because it tends to maintain a constant orientation in space. Inertial guidance systems are used in aircraft, spacecraft, oceangoing vessels, and missiles.

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Position Vector: The position vector locates any given point in the three dimensional rectangular coordinate system. Let us recollect that the three dimensional rectangular coordinate plane has x, y and z axes. The origin point for such coordinate system is given by (0,0,0). The position vector or any point P in space is given by OP (x, y, z). The magnitude of the vector |OP| can be given by √(x2+ y2 + z2).

So, we have |OP| = √(x2 + y2 + z2).

Let us take a look at one of such example question.

Question: What is the position vector of the point (3, 4, 5). Find the magnitude of the given position vector?

The position vector is found from origin (0, 0, 0) to the point P. Follow the steps in the diagram below for finding the magnitude of the given vector:

Let us now take a look at some basic types of vector spaces.

Null vector: A null vector is also called as the zero vector. Remember the starting point of the vector is called the initial point and the ending point is called as the terminal point. The vector for which both the initial point and terminal point are the same is called as the zero vector. Like the name suggests the magnitude of a zero vector is always equal to zero. The null vectors are written as PP, AA and so on.

Unit Vector: The magnitude of a vector when equal to 1 is called the unit vector.

Need an example? Here you go!

Find the magnitude of the vector AB given by (0, 1, 0).

Apply the formula for the magnitude of the vector. This is what you get:

Equal vector: Like the name suggests the equal vectors are the vectors which are the same in magnitude and direction. If AB and CD are two equal vectors, then they are given by AB = CD.

Negative Vector: Well again like the name suggests the vectors which have the same magnitude and are opposite in the direction are called the negative vectors. If OP is a vector than PO is called the negative vector. This can be represented as OP = – PO.

Here AB = – BA. So BA is the negative vector for vector AB.

Co-initial Vectors: Any given two vectors are called as co-initial vectors if both the given vectors have the same initial point. For example OA and OB are two co-initial vectors.

Collinear Vectors: Any two given vectors are called collinear vectors when both vectors are parallel to the same line.

In the above diagram AB and CD are parallel to the same line, hence the two vectors are collinear to each other.

Addition of vectors: Most students find it quite confusing to understand the addition of vectors. In simple terms here is how you can follow it:

In the above diagram E is the initial point for the vector EF and terminal point for vector DF. So to obtain the resultant we start from the initial point of one vector (DE) to the terminal point of another vector (EF). Hence the resultant of the vectors is DF.

                        This can be expressed as DF (resultant vector) = DE + EF.

Now that you are aware of the vector additions let us look at the properties of the vector space addition.

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