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.

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.

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.

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.

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.

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.

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).

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 coordinates of a point lying in the XY-plane will be of the form (x, y, 0).

The coordinates of a point lying in the YZ-plane will be of the form (0, y, z).

The coordinates of a point lying in the ZX-plane will be of the form (x, 0, z).

Let's now try and solve the following puzzle to check whether we have understood the basic concepts that we just studied.

Solved Examples

Example 1

State whether the following statements are true or false.

The point (−5, 0, 1) lies on the y-axis.

The point (1, 7, −1) lies in octant V, whereas the point (−10, −8, 6) lies in octant VIII.

The x, y, and z coordinates of the point (5, 7, 18) are 5, 7 and 18 respectively.

The point (0, 0, 19) lies in the ZX-plane.

The point (−2, 11, 8) lies in octant II.

Solution:

False. The point (−5, 0, 1) does not lie on the y-axis since the point that lies on the y-axis is of the form (0, y, 0).

False. Point (−10, −8, 6) lies in octant III.

True.

False. A point lying in the ZX-plane is of the form (x, 0, z).

True.

Example 2

Locate point (3, 2, −1) in a three-dimensional space.

Solution:

We have to locate point (3, 2, −1) in a three-dimensional space. In order to do this, we will first draw the three axes.

Then, starting from the origin, when we move 2 units in the positive y-direction and then 3 units in the positive x-direction, we will reach at the point A(3, 2, 0).

From point A(3, 2, 0), we move 1 unit in the negative z-direction to reach the required point i.e., P(3, 2, −1).

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.

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.

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.

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.

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.

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.

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…

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