In three-dimensional geometry, direction ratios are used to represent the direction of a straight line in space. A direction ratio is a scalar value that specifies the change in the coordinates of a point on the line with respect to a unit change in the distance along the line in a particular direction. These direction ratios are used to find the direction cosines, which are normalized direction ratios, with a magnitude of 1.
Let’s consider a line passing through two points P(x1, y1, z1) and Q(x2, y2, z2) in three-dimensional space. The direction ratios of this line can be calculated as follows:
Direction Ratio in x-direction = (x2 – x1)
Direction Ratio in y-direction = (y2 – y1)
Direction Ratio in z-direction = (z2 – z1)
Once you have the direction ratios, you can find the direction cosines by dividing each direction ratio by the magnitude of the line segment between points P and Q.
For example, consider two points A(1, 2, 3) and B(4, 5, 6). The direction ratios of the line passing through these points are:
Direction Ratio in x-direction = (4 – 1) = 3
Direction Ratio in y-direction = (5 – 2) = 3
Direction Ratio in z-direction = (6 – 3) = 3
Now, to find the direction cosines, first, calculate the magnitude of the line segment AB:
Magnitude AB = √((4 – 1)^2 + (5 – 2)^2 + (6 – 3)^2) = √(3^2 + 3^2 + 3^2) = √27
Now, divide each direction ratio by the magnitude:
Direction Cosine in x-direction = 3 / √27 ≈ 0.577
Direction Cosine in y-direction = 3 / √27 ≈ 0.577
Direction Cosine in z-direction = 3 / √27 ≈ 0.577
The direction cosines give us the direction of the line in terms of unit vectors. In this example, the line passing through points A and B has the direction of the vector (0.577i + 0.577j + 0.577k). This means that the line is inclined at equal angles with respect to the x, y, and z-axes.
Direction ratios are essential in three-dimensional geometry as they help define the orientation of lines, planes, and vectors in space, making it easier to understand their spatial relationships and perform various calculations involving directions and angles.
Key for ATOQ (Any Time Online Quiz)
- The direction ratios of a line passing through points P(2, 4, 6) and Q(1, 3, 5) are -1, -1, -1.
- For a line with direction cosines (0.6, 0.8, 0), the direction ratios are 0.6, 0.8, 0.
- The direction ratios of a line perpendicular to the line with direction cosines (1, 2, -2) are 2, -1, 1.
- Find the direction cosines of a line with direction ratios (3, 6, 9): 3/√126, 6/√126, 9/√126.
- A line with direction cosines (0, -1, 0) lies along the y-axis.
- If the direction cosines of a line are (0.707, -0.707, 0), the direction ratios are 0.707, -0.707, 0.
- The direction cosines of a line parallel to the xy-plane are (0, 0, 1).
- The direction ratios of a line passing through points A(3, 2, 1) and B(-1, 0, -2) are -4, -2, -3.
- For a line with direction ratios (5, 2, -3), the direction cosines are 5/√38, 2/√38, -3/√38.
- The direction ratios of a line parallel to the xz-plane are 2, 0, -1.
- Find the direction cosines of a line with direction ratios (2, 4, 2): 2/√24, 4/√24, 2/√24.
- The direction cosines of a line perpendicular to the z-axis are (0, 0, 1).
- The direction ratios of a line passing through points M(2, -1, 5) and N(3, 1, 4) are 1, 2, -1.
- If the direction ratios of a line are (-2, 1, -5), the direction cosines are -2/√30, 1/√30, -5/√30.
- A line with direction cosines (1, 0, 0) lies along the x-axis.
- The direction cosines of a line parallel to the xy-plane are (1, 0, 0).
- The direction ratios of a line passing through points X(4, 1, 0) and Y(2, 3, -1) are -2, 2, -1.
- For a line with direction ratios (0, -3, -4), the direction cosines are 0, -3/√25, -4/√25.
- The direction cosines of a line perpendicular to the yz-plane are (1, 0, 0).
- Find the direction ratios of a line with direction cosines (0, 0.6, -0.8): 0, 0.6/√1.00, -0.8/√1.00.