Ch 5Vectors — Class 12 Mathematics, concept-first.

Many physical quantities can be sorted into two kinds. A scalar quantity is completely described by its magnitude alone -- mass, length, temperature, area, volume, time, distance, speed, work, money,…

A vector is drawn as a directed line segment: if and are two points and the segment carries an arrowhead at , the resulting directed segment represents the vector (read "AB bar" or "vector AB"), with…

The magnitude (also called the size or length) of a vector is written and is defined simply as the length of the segment , i.e. . If vectors are written as , their magnitudes are written .

Vectors are sorted into several named types by special features of their magnitude, direction or initial point.

Scalar multiplication. For a vector and a real number (scalar) , the scalar multiple is the vector with magnitude , pointing in the same direction as when and in the opposite direction when ; when , .…

If and are two vectors, their sum (resultant) can be constructed by either of two equivalent laws.

Subtraction has already been introduced above as : to subtract from , reverse 's arrow and apply the triangle law. In coordinate form this is simply subtracting like components.

Two or more vectors are coplanar if they can be drawn lying in a single plane, or in mutually parallel planes.

Restricting to a single plane, the set of all combinations (for real ) of two non-collinear vectors sweeps out the entire plane; are then called the generators of that plane, and its basis.

A single point in a plane needs only an ordered pair (its signed distances from the Y- and X-axes). Locating a point in space, however, needs three numbers: three mutually perpendicular coordinate axe…

To describe a vector's direction precisely, it is written as a linear combination of three special basis vectors.

For a point in space (with respect to the origin ), the vector is called the position vector of with respect to .

If is the position vector of a point with respect to the origin, then , and are called the components of along the axes; every vector in space is a unique combination of this way (some texts write thi…

If is the origin and are the position vectors of , then in , using the triangle law, i.e. the vector joining two points is (position vector of the end point) minus (position vector of the start point)…

Theorem 5 (Section formula, internal division). Let and be two points and a point on segment dividing it internally in the ratio (so -- in order and ).

Two non-zero vectors , drawn from a common initial point, make an angle with ; this angle is written . Collinear vectors make an angle of if they point the same way and if opposite.

Rearranging the dot-product definition, the angle () between two non-zero vectors is Since cosine is positive for , zero at , and negative for , the sign of alone already tells whether the angle betwe…

Let and share initial point , and let be the foot of the perpendicular from to the line containing (so , with , is that foot). Then:

For a non-zero vector , its direction angles are the angles it makes with the positive axes, and their cosines are its direction cosines (d.c.'s). Since is the angle between and , , and similarly .

To describe how a plane is tilted (rather than just a line, as slope does), two vectors lying in the plane are combined to produce a third vector perpendicular to that plane -- this is the vector (cro…

Definition. For three vectors (order matters), the scalar triple product is , written ; for etc., It is also called the box product.

For vectors , the vector triple product is itself a vector (not a scalar, despite superficially resembling the scalar triple product), and is stated (without proof at this level) to equal

Sample papers and previous-year board questions for this subject.