. . Geometrical interpretation Geometrically, if a is an arbitrary vector and ˆ n is a unit vector, then a n is the projection of the vector a on the straight line on which ˆ n lies. The quantity a n is positive if the angle between a and ˆ n is acute, see Fig.
. and negative if the angle between a and ˆ n is obtuse see Fig. . .
If a and b are arbitrary non-zero vectors, then | a b b a a b ⋅ = ⋅ and so a b means either the length of the straight line segment obtained by projecting the vector | b a along the direction of b or the length of the line segment obtained by projecting the vector | a b along the direction of a . We recall that | | | cos a b , where θ is the angle between the two vectors a and b . We recall that the angle between a and b is defined as the measure from a to b in the counter clockwise direction. The vector a is either or a vector perpendicular to the plane parallel to both a and b having magnitude as the area of the parallelogram formed by coterminus vectors parallel to a and .
If a and b are non-zero vectors, then the magnitude of a can be calculated by the formula = | | | | | sin |, where θ is the angle between a and b Two vectors are said to be coterminus if they have same initial point. ˆ n a n q Positive dot product a n ˆ n Negative dot product Vector - - Applications of Vector Algebra Remark ( ) An angle between two non-zero vectors a and b is found by the following formula | | a b − ( ) a and b are said to be parallel if the angle between them is or π . ( ) a and b are said to be perpendicular if the angle between them is π or p . Property ( ) Let a and b be any two nonzero vectors.
Then a b if and only if a and b are perpendicular to each other. if and only if a and b are parallel to each other. ( ) If , a b , and c are any three vectors and α is a scalar, then a b b a a b b c a b , ( , ( ); = − ), ( , (