Vectors and 3 Dimensional Geometry Simplified for Class 12
A quantity which has only magnitude and no direction is called a scalar. A quantity which has both magnitude and direction is called a vector. A portion of a line segment whose two endpoints are distinguished as initial and terminal is called a directed line segment.
You'll need to understand what is meant by position vector of a point. In this context, keep in mind the terms inital point and terminal point of a vector. Next, you'll learn about direction ratios and direction cosines of a vector. Direction cosines of a vector can also be defined as the angle made by a vector with the positive directions of the x, y and z axes respectively.
A vector whose magnitude is zero is called a null vector. A vector whose magnitude is one is called a unit vector. Two vector can be added using the Triangle law.As a sequel to this, you have the parallelogram law of addition of vectors.
Next, you'll move on to dot product or Scalar product of two vectors. This definition makes use of the angle between two vectors. Note that the dot product of two vectors is a scalar. If two vectors are perpendicular to each other, their dot product is zero.
In this context, you'll learn about the modulus or magnitude of a vector which is the square root of the dot product of a vector with itself. There are also important results about the dot products of unit vectors along the coordinate axes. A very important result that you will need to learn is how to calculate the projection of one vector on another vector.
Scalar product is commutative, associative and is distributive over addition. You'll need to learn how to calculate the dot product of two vectors, just given their rectangular coordinates.
As you progress, you'll learn about Vector Product and Cross Product of two vectors. Keep in mind that the vector product is a vector and it's direction is given by the Right Hand Screw Rule. A unit vector is perpendicular to two vectors a and b if it is perpendicular to the plane of a and b. Hence Cross product is used to find a unit vector perpendicular to two vectors.
If the cross product of two vectors is zero, the vectors are perpendicular to each other. Hence, cross product of a vector with itself is zero. Vector product is not commutative.
The three coordinate exes form a right handed triad of mutually perpendicular unit vectors. The vector product is distributive over addition. You'll learn how to calculate the cross product of two vectors using Determinants.
Yet another extremely important identity which you'll need to learn is Lagrange's Identity. This identity connects the scalar product with cross product. You can also use Cross product to find the area of a parallelogram, given it's two adjacent sides and the area of a triangle given it's adjacent sides. You can proceed one step further and use Cross product to find the area of a parallelogram given it's diagonals.
Lastly, we move on to Scalar Triple Product. The scalar triple product is basically the dot product of two vectors, one of which is a vector product of two vectors. The scalar triple product is always a scalar.
An important point to note is that the scalar triple product of three vectors is the volume of the parallelopiped with the three vectors as the coterminus edges.You can calculate the scalar triple product by expanding a three by three determinant where each vector is a row of the determinant. Three vectors are coplanar if their scalar triple product is zero.
The value of the scalar triple product is unchanged as long as the cyclic order of the vectors is maintained. If the cyclic order is broken, the value of the scalar triple product changes in sign. If any two of the given vectors are equal, the value of the scalar triple product becomes zero. Again, if any two vectors are collinear or parallel, the scalar triple product becomes zero
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Three Dimensional Geometry for Class 12 Mathematics
Welcome to three dimensional Geometry for Class 12. I am Suman Mathews, mathematics teacher and content developer with a teaching experience of three decades. Students in general tend to find this topic difficult. I hope I can help eliminate this difficutly for you. Following are the topics which I will be teaching in my online classes along with lots and lots of problem solving.
You'll start with by learning the direction ratios and direction cosines of a line. The direction ratios are the angles made by a line with the positive directions of the x,y and z axes respectively. The direction cosines are the cosines of these angles.
Learn the formulas for finding the direction ratios and direction cosines of a line given two points on a line. Using this, you can also find the angle between the two lines. Note that when you take the modulus, you'll get the acute angle between the two lines. If the cosine of the angle between the lines is zero, the lines are perpendicular. Yet another important point to keep in mind is that when two lines are parallel, their direction ratios are proportional.
Moving on to equation of a line in three dimensional geometry. You need to learn how to write the equation of a line in Cartesian and Vector form. Basically, you should be able to convert one form to the other.
You'll need to learn how to calculate the equation of a line passing through a given point and parallel to a given vector. Equation of a line passing through two points is also what you'll be learning.The Cartesian form of a line is slightly easier to calculate and then you can always convert to the Vector form.
An important aspect of three dimensional Geometry that you'll need to know is finding the angle between two lines. This can be done using the vector form or Cartesian form of a line. Note that when you apply the modulus to the formula, it's finding the acute angle between two lines.
You'll also need to know how to find the equation of a line passing through a point and perpendicular to two lines.
Next, you'll need to learn how to calculate the shortest distance between two lines. The vector form of the equation is easier to calculate in this case. Keep in mind that two non parallel lines intersect if and only if the shortest distance between them is zero. You can also calculate the shortest distance between two parallel lines as an application of this formula.
A plane is a surface such that if any two distinct points are taken on it, then the line containing these points lies completely in it. You'll start with calculating the equation of a plane perpendicular to a given direction and at a distance p from the origin. This is the normal form and all other equations can be derived from this.
You'll also learn how to calculate the equation of a plane perpendicular to a given direction and passing through a given point. Again, you can write the equation of the plane in the Cartesian and Vector form. You should know how to derive one form from the other. All these problems involve identifying the direction ratios of the normal to the plane.
The next section is extremely important as it covers various points extremely important in three dimensional geometry. These include
Finding the equation of a plane passing through two given points and parallel to a given line.
Finding the equation of a plane passing through a point and parallel to two non parallel lines.
Finding the condition of coplanarity of two lines.
Finding the equation of a plane containing two lines.
Finding the equation of a plane passing through three points.
Intercept form of a plane.
Finding the equation of a plane passing through the intersection of two planes. This is essentially the most important part of planes and you'll need to have a good understanding of these. I would personally recommend using the Cartesian form to calculate each of the above. You simply have to reduce everything to a 3 by 3 determinant.
Moving on, you'll learn how to calculate the angle between two planes. Calculating the angle between a line and a plane is also important and something that you have to know.
How to calculate the distance of a point from a plane is what you'll learn next. That means you'll calculate the length of the perpendicular from a point to a plane. As I mentioned earlier, it's easier to remember the Cartesian form of the formula here. To calculate the perpendicular distance between two parallel planes, take any point on one plane and find the perpendicular distance from this point to the other plane.
Learn all this and more by registering for my online classes. You'll learn multiple choice questions, Assertion based questions, case study based questions, conceptual based questions and more. There will also be tests at the end of every chapter.
You can sign up for questions on Three dimensional geometry. So, let's start the learning process.