Question

The equation of line passes through \((1,2,3)\) and \((x / 1)=(y / 2)=[z /(-1)]\) and \([(x-1) / 3]=(y / 2)=(z / 6)\) and perpendicular to given two line is (A) \([(\mathrm{x}-1) / 14]=[(\mathrm{y}-2) /(-9)]=[(\mathrm{z}-3) /(-4)]\) (B) \([(\mathrm{x}+1) / 14]=[(\mathrm{y}+2) /(-9)]=[(\mathrm{z}+3) / 4]\) (C) \([(\mathrm{x}-1) / 14]=[(\mathrm{y}-2) / 9]=[(\mathrm{z}-3) /(-4)]\) (D) \([(\mathrm{x}+1) /(-14)]=[(\mathrm{y}+2) / 9]=[(\mathrm{z}+3) / 4]\)

Short answer

The equation of the required line is: \([(x-1)/4]=[(y-2)/(-9)] =[(z-3)/0]\) None of the given options are correct.

Step-by-step solution

Find the Direction Ratios of the Given Lines

For the first line, let's compare the given equation with \((x-x_1) / a_1 = (y-y_1) / b_1 = (z-z_1) / c_1\). We have: - For line 1: \((x-1)/1 = y/2 = z/-1\) Direction ratios are \(a_1=1, b_1=2,\) and \(c_1=-1\) - For line 2: \((x-1)/3 = y/2 = z/6\) Direction ratios are \(a_2=3, b_2=6,\) and \(c_2=6\)

Find the Direction Ratios of the Line Perpendicular to Both Given Lines

To find the direction ratios of a line perpendicular to both given lines, we will take the cross product of their direction ratios: \(\textbf{P} = (1,2,-1) \times (3,6,6)\) \(\textbf{P} = \langle 2(-1)-(-1)(6), (-1)(3)-(1)(6), (1)(6)-2(3) \rangle = \langle -2+6, -3-6, 6-6 \rangle = \langle 4, -9, 0 \rangle\) So, the direction ratios of the line perpendicular to both given lines are: \(a=4, b=-9,\) and \(c=0\)

Form the Equation of the Required Line

Now, let's form the equation of the required line using the point \((1,2,3)\) and the direction ratios \(a=4, b=-9,\) and \(c=0\). We have: \((x-1) / 4 = (y-2) / -9 = (z-3) / 0\) Since the value of \(c\) is zero, the third component determines that the line passes through point \((1,2,3)\). The equation of the required line is: \([(x-1)/4]=[(y-2)/(-9)] =[(z-3)/0]\) However, none of the given options present an equation identical to our derived equation. Comparing the given options with our derived equation: (A) is incorrect since the third direction ratio \((z-3)/(-4)\) does not match our equation. (B) contains different directions ratios and wrong signs in the numerators. (C) contains correct direction ratios but a wrong sign in the denominator of the second term. (D) contains wrong signs in the equation and different direction ratios. Therefore, none of the given options are correct.

Key concepts

Direction Ratios

When working with lines in 3D coordinate geometry, understanding direction ratios is crucial. These numbers provide a way to describe the direction in which a line extends into three-dimensional space. They are similar to the concept of the slope in 2D, but since we're in 3D space, we need three values instead, one for each axis (x, y, z).

In the context of the given exercise, direction ratios were extracted by comparing the equation of a line \( (x-x_1) / a_1 = (y-y_1) / b_1 = (z-z_1) / c_1 \) with the given lines. These ratios serve as coefficients that scale the unit direction along the respective axes, essentially giving us the components of a direction vector for the line.

By finding the direction ratios of two lines, we can proceed to determine the equation of a third line that might be parallel, perpendicular or at some other specific angle to the original lines. This step forms the foundation for solving the given question in the exercise.

Line Perpendicular to Two Lines

Finding a line that is perpendicular to two given lines involves a bit of vector mathematics, particularly using the cross product. The cross product is a special operation that takes in two vectors and returns a third vector which is perpendicular to the plane formed by the initial two.

In three dimensions, for two vectors \( \textbf{A} \) and \( \textbf{B} \) given by their direction ratios, the cross product \( \textbf{A} \times \textbf{B} \) gives us the direction ratios of a new vector \( \textbf{P} \) that is perpendicular to both \( \textbf{A} \) and \( \textbf{B} \) .

This operation is crucial in our exercise to find the direction ratios for the line that is perpendicular to the two provided lines. By applying the cross product to the direction ratios of the given lines, we can find a new set of direction ratios, which can then be used to form the equation of a line that perfectly meets the condition of being perpendicular to both.

Cross Product in Vector

The cross product is an operation in vector algebra that returns a vector perpendicular to the original two vectors. If we consider vectors in the 3D space as \( A = (a_1, b_1, c_1) \) and \( B = (a_2, b_2, c_2) \), the cross product \( A \times B \) is given by:

\[ \textbf{C} = (b_1c_2 - c_1b_2, c_1a_2 - a_1c_2, a_1b_2 - b_1a_2) \]
This result \( \textbf{C} \) is a vector that is orthogonal to both \( A \) and \( B \). In our problem, this operation was used to ascertain the direction ratios of the line perpendicular to both given lines. The correctness of the cross product is verified using the fact that \( A \) and \( B \) have zero dot product with \( C \) because they are perpendicular.

3D Coordinate Geometry

3D coordinate geometry involves the study of points, lines, and planes in three-dimensional space. Lines in 3D are represented by equations that give us the set of all points that lie on the line. Unlike 2D, where a line's equation is often written in slope-intercept or point-slope form, a 3D line requires a vector approach, usually involving direction ratios and a given point on the line.

The general form of the equation of a line in 3D is \( (x-x_1) / a = (y-y_1) / b = (z-z_1) / c \), where \( (x_1, y_1, z_1) \) is a point through which the line passes, and \( a, b, c \) are direction ratios representing the direction of the line. In many exercises, including the one in discussion, determining the equation of a line involves finding these parameters, so it adheres to the conditions such as passing through a particular point or being perpendicular to other lines.