A. The Equation of a Straight Line VECTORS P3 VECTORS IN A STRAIGHT LINE A particular line is uniquely located in space if : I. It has a known direction, d, and passed through a known fixed point, or II. It passes through two known fixed points. I. A line with known direction, d, passing through a fixed point d A The diagram shows a straight line passing through a fixed point with a position vector a which is parallel to a given vector d. a r R Let r be the position vector of a point R on the line. O Since AR is parallel to d, then: r a AR = d = d r = a + d Thus every point on the line has a position vector of the form a + d. The equation r = a + d is called the vector equation of the line through A parallel to d. The vector d is a direction vector of the line. Examples :. Find the vector equation of the line which is parallel to the vector i j + 3k and passes through the point 5 4. Hence write down the parametric and Cartesian equations of the line.
. Find the vector equation of the line passing through the point with position vector i j + 3k and parallel to the vector j k. Hence write down the parametric and Cartesian equations of the line. II. A line through two fixed points Consider now the line through two points A and B with position vectors a and b respectively. R A B a b r If a point R, with position vector r, lies on the line, then for some scalar λ, 0 Example : AR = λ AB r a = λ b a r = a + λ b a Find the vector equation of the straight line passing through the points A B 3 0 and. Find the parametric and Cartesian equations of the line. Find in each case the coordinates of the points where the line crosses the xy plane, the yz plane and the zx plane.
* A general point on the line is the parametric equation of the line. Example : x = + λ, y = 3λ, z = λ At the point where the line crosses the xy plane z = 0 3 Therefore the line crosses the xy plane at the point. Note : Crosses the xy plane Crosses the yz plane Crosses the xz plane B. Pairs of Lines The location of two lines in space may be such that I. The lines are parallel. II. The lines are not parallel and they intersect. III. The lines are not parallel and do not intersect. Such lines are called skew.. Parallel Lines If two lines are parallel, this property can be observed from their equations (such that they will have parallel direction vectors). Example : l : r = + λ 3 l : r = 4 + λ 6
. Non-parallel lines r = a + μd Direction ratios (d and d ) are different. Should the lines intersect, there must be unique values λ and μ such that : 4 r = a + λd r = r a + λd = a + μd Example: l : r = λ i + 3λj + λ + 5 k l : r = μi + μ 4 j + 3k If no such values can be found, then the lines do not intersect. 3. Skew Lines The lines are not parallel and do not have a point of intersection. We cannot find values for λ and μ which satisfy all three equations. Example: l : r = + λ i + 3λj + + 4λ k l : r = + 4μ i + 3 μ j + μk
C. Angle between a Pair of Lines The angle between two lines l and l is ambiguous as it may be α or 80 α. l But the angle between two vectors a and b is unique. 5 d 80 α It is the angle between their directions when those directions both converge or both diverge from a point. θ d α l If two lines have equations : l : r = a + λd and l : r = a + μd The angle between any two lines depends only on their directions : d. d = d d cos θ If the lines are perpendicular then, Example : Find the angle between the lines r = i j + 3k + λ i 3j + 6k r = i 7j + 0k + μ i + j + k
D. Perpendicular Distance from a Point to the Given Line In general P PQ is perpendicular to l or PQ is perpendicular to d. 6 Q d l Examples :. Find the perpendicular distance from a point P, with position vector with vector equation r = 3 + λ 3. 4 3 0 to the line l
. Find the perpendicular distance (shortest distance) of a point from a point A,, 3 7 to the line x 4 = y = z.
E. Vector Equations of a Plane Scalar Form n VECTORS IN A PLANE Consider the plane which contains the point A with position vector a and is perpendicular to n. If r is the positive vector for any point R on the plane. 8 A R Since AR is perpendicular to n. a r O Scalar Form : If r = xi + yj + zk n = Ai + Bj + Ck Cartesian Form Parametric Form q A p R Consider the plane is parallel to vector p and q (p is not parallel to q) and also contains point A with position vector a. If R is any point on the plane: a r O
Examples :. Find the vector equation of the plane passing through the point with position vector i j + k and is perpendicular to the vector 3i + j 4k. 9. Write down a vector equation of the plane which contains A,, 3 and is perpendicular to 3i + 4j + 5k in scalar and Cartesian form. Hence show that the point E, 5, 6 does not lie on the plane. 3. Find the vector equation of the plane that contains the three points, 5, 3,, 3, 5 and,, 0. Show that the point 0, 3, 0 lies on the plane.
4. Find the Cartesian equation from the vector equation of a plane : r = 5 3 + λ 3 + μ 4 3 0 5. Find the Cartesian equation of the plane that passes through the points,,, 3,, and,,.
6. Find the Cartesian equation of the plane through the point, 3, parallel to the plane x + 4y 5z =. F. Intersection of a line and a plane To find where a line meets a plane, we need to find a point on the plane that satisfies the equation for the plane. Steps :. Write down the coordination of a general point on the line.. Use these coordinates in the formula for the equation of the plane. 3. Solve the equation and find the coordinates. Examples :. Find where the line = 0 + 4 meets the plane + =.. Find the vector equation of the line passing through the point 3,, and perpendicular to the plane. + = 4. Find also the point of intersection of this line and the plane.
G. To Prove that a Line Lies in a Plane Conditions :. The line must be parallel to the plane.. The point through which the line passes must satisfy the plane. Examples :. Show that the line = + + + lies in the plane. 3 + = H. The Angle Between Two Planes The angle between the normals is :. = cos Example : Find the acute angle between two planes whose vector equation are. + = 3 and. + =
I. The Angle Between a Line and a Plane 3 Example : Find the angle between the line = + + + and the plane. + = 4 The Line Intersection of Two Planes ( When two planes intersect, they form a line) As the line of intersection of two planes. =. = Is contained in both planes, it is perpendicular to both and.
4. Find the line of intersection of these two planes in both Cartesian and Vector equation;. + 3 = 6 Examples :. + = 4
. Find in both Cartesian and vector equation of the line of intersection of the two planes: 7 4 + 3 = 3 4 + + = 4 5
J. The Distance of a Point from a Plane The Distance of a point P from a plane 6. = is. where = And assuming that the point P lies on the plane which is parallel to, then therefore the equation of plane Where distance from the origin =... =. Thus the distance of a point P from a plane is Note :. If P and O (origin) are on the same side of plane, the result will be negative (-ve).. If P and O (origin) are in the opposite side of the plane, the result will be positive (+ve). Example : Find the distance of the point, 3, 3 from a the plane with equation. + 3 6 = 9