Vectors and 3 Dimensional Geometry Simplified for Class 12
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Vector Formulas Part 1 Understanding the Mysteries of Magnitude and Direction
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.
MCQ-Finding a vector in the direction of another vector
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.
MCQ-Length of the diagonal of a rectangular parallelopiped
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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Discover Formulas in Vectors for Class 12