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Vectors

Vector and its Related Concepts

  • If we draw three planes intersecting at O such that they are mutually perpendicular to each other, then these will intersect along the lines X′OX, Y′OY and Z′OZ. These lines constitute the rectangular coordinate system and are respectively known as the x, y, and z-axes.

    • Point O is called the origin of the coordinate system.

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  • The distances measured from XY-plane upwards in the direction of OZ are taken as positive and those measured downward in the direction of OZ′ are taken as negative.

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  • The distances measured to the right of ZX-plane along OY are taken as positive and those measured to the left of ZX-plane along OY′ are taken as negative.

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  • The distances measured in front of YZ-plane along OX are taken as positive and those measured at the back of YZ-plane along OX′ are taken as negative.

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  • The planes XOY, YOZ, and ZOX are known as the three coordinate planes and are respectively called the XY-plane, the YZ-plane, and the ZX-plane.

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  • The three coordinate planes divide the space into eight parts known as octants. These octants are named as XOYZ, X′OYZ, X′ OY′Z, XOY′Z, XOYZ′, X′OYZ′, X′OY′Z′, and XOY′Z′ and are denoted by I, II, III, IV, V, VI, VII, and VIII respectively.

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  • If a point A lies in the first octant of a coordinate space, then the lengths of the perpendiculars drawn from point A to the planes XY, YZ and ZX are represented by x, y, and z respectively and are called the coordinatesof point A. This means that the coordinates of point A are (x, y, z). However, if point A would have been in any other quadrant, then the signs of x, y, and z would change accordingly.

  • The coordinates of the origin are (0, 0, 0).

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  • The sign of the coordinates of a point determines the octant in which the point lies. The following table shows the signs of the coordinates in the eight octants.

Octants

I

II

III

IV

V

VI

VII

VIII

Coordinates↓

x

+

+

+

+

y

+

+

+

+

z

+

+

+

+

  • The coordinates of a point lying on different axes are as follows:

    • The coordinates of a point lying on the x-axis will be of the form (x, 0, 0).

    • The coordinates of a point lying on the y-axis will be of the form (0, y, 0).

    • The coordinates of a point lying on the z-axis will be of the form (0, 0, z).

  • The coordinates of a point lying on different planes are as follows:

    • The coo…

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