The projection of a vector along an axis of a coordinate system is called a "component" of the vector. For a given vector, its component along a specific axis is determined by taking the dot product of the vector with a unit vector in the direction of that axis. This process effectively measures how much of the vector aligns with that axis. Each axis in the coordinate system has its own corresponding component of the vector.
If you have a vector of magnitude r, making an angle of a degrees, then its projection on the x-axis is r*cos(a) and on the y-axis it is r*sin(a).
I suspect the question arises from confusion. A vector itself already defines a direction, usually in the Cartesian xyz coordinate system. If you want to express the direction in other coordinates, such as polar or spherical coordinates you need to transform the vector to these coordinate systems. I can answer you question more fully if you can specify the specific coordinate system in which you want to know the direction.
Ans :The Projections Of A Vector And Vector Components Can Be Equal If And Only If The Axes Are Perpendicular .
Components.
You get other vectors, usually perpendicular to each other, that - when added together - result in the original vector. These component vectors are usually along the axes of some selected coordinate system.
A tangent of the vector is the projection of a vector along the axes of a coordinate system.
Yes, that is correct. The components of a vector, which represent its magnitude and direction in a particular coordinate system, are independent of the choice of coordinate system used to express the vector. This property is a fundamental characteristic of vectors in mathematics and physics.
If you have a vector of magnitude r, making an angle of a degrees, then its projection on the x-axis is r*cos(a) and on the y-axis it is r*sin(a).
No, there are multiple coordinate systems in which vector components can be added, such as Cartesian, polar, and spherical coordinates. The choice of coordinate system depends on the problem at hand and the geometry of the situation being analyzed.
The Cartesian coordinates of the vector represented by the keyword "r vector" are the x, y, and z components of the vector in a three-dimensional coordinate system.
I suspect the question arises from confusion. A vector itself already defines a direction, usually in the Cartesian xyz coordinate system. If you want to express the direction in other coordinates, such as polar or spherical coordinates you need to transform the vector to these coordinate systems. I can answer you question more fully if you can specify the specific coordinate system in which you want to know the direction.
The zero vector has no direction because it has a magnitude of zero. It is represented by a point at the origin in a coordinate system, with no specific direction.
The length of a vector is a scalar quantity, typically denoted as a positive real number, that represents the magnitude or size of the vector. It is calculated using the vector's components in a coordinate system, often with the Pythagorean theorem.
Ans :The Projections Of A Vector And Vector Components Can Be Equal If And Only If The Axes Are Perpendicular .
The magnitude of a vector remains the same across different coordinate systems, regardless of the orientation or direction of the vector.
To show the correct direction of a vector, you need to specify the reference point or origin from which the vector is being measured, and also indicate the angle or orientation at which the vector is pointing relative to that reference point. This information can be represented using coordinate axes, angles, or directional headings.
In a given coordinate system, the components of a vector represent its magnitude and direction along each axis. Unit vectors are vectors with a magnitude of 1 that point along each axis. The relationship between the components of a vector and the unit vectors is that the components of a vector can be expressed as a combination of the unit vectors multiplied by their respective magnitudes.