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Maths > Three Dimensional Geometry > Equation of Line for Space
Three Dimensional Geometry

Equation of Line for Space

The equation of a line in a plane is given by the popular equation y = m x + C. We must, however, look at how the equation of a line is written in vector form and Cartesian form. This lesson equation of line explains how the equation of a line in 3-D space can be found. A line is said to be unique if it passes through a given point and has a direction or if it passes through two given points. Let us also study the equation of a straight line.

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Vector Equation

Let us consider a line that passes through a given point, say A, and the line is parallel to a given vector \( \vec{b} \). Here, the line l is given to pass through A, whose position vector is given by \(\vec{a}\).  Now let us consider another arbitrary point P on the given line, where the position vector of P is given by \(\vec{r}\).

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Since \( \vec{AP} \) is parallel to the vector \(\vec{b} \), we write as

$$ \vec{AP} = \lambda \vec{b} $$

But we also know that the vector \vec{AP} can be written as

$$ \vec{AP} = \vec{OP} – \vec{OA} $$

$$ \lambda \vec{b} = \vec{r} – \vec{a} $$

Rearranging the equation, we have

$$ \vec{a} = \vec{a} +  \lambda \vec{b} $$

Note that \( \vec{b} \) may be given to you in the form as –

$$ \vec{b} = b_1 \hat{i} + b_2 \hat{j} + b_3 \hat{k} $$

Here, \( b_1 , b_2 \;  and \; b_3 \) are the direction ratios of the vector \(\vec{b}\).

Now let us proceed to the Cartesian equation of the line in space.

Straight line

Source: Youtube

Cartesian Equation

The Cartesian equation of a line in space can be explained in a similar manner. Let the coordinates of a point A through which the line passes are \( (x_1, y_1, z_1) \) and the direction ratios of the line be a, b, c. So  we write the equation as

$$ \vec{r} = x\hat{i} + y\hat{j} + z\hat{k} $$

$$ \vec{a} = x_1 \hat{i} + y_1 \hat{j} + z_1 \hat{k} $$

and, $$ \vec{b} = a \hat{i} + b \hat{j} + c \hat{k}  $$

Now, substituting in the vector form of the equation of a line –

$$ x\hat{i} + y\hat{j} + z\hat{k} = x_1 \hat{i} + y_1 \hat{j} + z_1 \hat{k} + \lambda (a \hat{i} + b \hat{j} + c \hat{k} ) $$

$$ x\hat{i} + y\hat{j} + z\hat{k} = ( x_1 + `\lambda a ) \hat{i} + (y_1 + `\lambda b ) \hat{j} + ( z_1 + `\lambda c ) \hat{k} $$

So now we have –

$$ x = x_1 + `\lambda a $$

$$ y = y_1 + `\lambda b $$

and,

$$ z = z_1 + `\lambda c $$

Solving for \( \lambda \) in each equation gives us the equation of the straight line by equating all the respective values of \( \lambda \)  –

$$ \frac{x – x_1}{a} = \frac{y – y_1}{b} = \frac{z – z_1}{c} $$

This is the equation of a straight line in space in Cartesian form. The following section contains a solved question for you to understand how problems on this topic can be tackled step by step.

Solved Example for You on Straight Line

Question 1: Find the Cartesian equation of the line that passes through the point A (1, 2, 1) and whose direction vector is given by (4, 5, -1)

Answer: We can find the equation of the line in Cartesian form by using the formula above as –

\( x_1 = 1 , y_1 = 2, z_1 = 1 \; and \; a = 4, b = 5, c = -1 \)

Thus we can write the equation as –

$$ \frac{x – 1}{4} = \frac{y – 2}{5} = \frac{z – 1}{-1} $$

Question 2: Explain what is a straight line in math.

Answer: We can define a straight line as the set of all points between and extending beyond two points. In addition, two properties of straight lines in Euclidean geometry are that they have only one dimension, length, and they extend in two directions forever.

Question 3: How to calculate a straight line?

Answer: For calculating straight line the general equation is y = mx + c, where m is the gradient, and y = c is the value where the line cuts the y-axis. In addition, the value of c or number c is known as the intercept on the y-axis. Moreover, the equation of a straight line with gradient m and intercept c on the y-axis is y = mx + c.

Question 4: How to find the slope of a straight line?

Answer: The slope of a line illustrates the course of a line. For finding the slope you divide the difference of the y-coordinates of 2 points on a line by the difference of the x-coordinates of those same 2 points.

Question 5: What is a vertical line?

Answer: It refers to a line on the coordinate plane where all points on the line have the same x-coordinate.

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