Definition: Consider the two vectors and . If the product of two vectors and is a scalar then the multiplication is called scalar product (or) dot product. It is written as . . It is read as dot and it is defined as
Where (a, b) means the angle between and
Note 1. If a = a1i + a2 j + a3k and b = b1i + b2j + b3k. Then
a.b = a1b1 + a2b2 + a3b3
2. If i, j, k are orthogonal unit vector triad in a right handed system, then
3. If = a1i + a2j + a3k and = b1i + b2j + b3k are any two perpendicular vectors then, a1b1 + a2b2 + a3b3 = 0
4. If = a1i + a2 j + a3k and = b1i + b2j + b3k are parallel vectors then
5. a2 = a.a
6. a.b = b.a
7. (a + b)2 = a2 + b2 + 2 a.b
8. (a - b)2 = a2 + b2 - 2 a.b
9. (a + b + c)2 = a2 + b2 + c2 + 2 a.b + 2 b.c + 2 c.a
10. (a + b)2 + (a - b)2 = 2(a2 + b2)
11. (a + b)2 - (a - b)2 = 4 a.b
12. a2 - b2 = (a + b).(a - b)
Conceptual Theorems
1. If is any vector then = (r.i) i + (r.j) j + (r.k) k
Proof: Let = xi + yj + zk
r.i = (xi + yj + zk).i
r.i = x(i.i) + y (j.i) + z (k.i)
r.i = x(1) + y(0) + z(0) [ from Note (2) ]
r.i = x
(r.i)i = xi ........................ (1)
Similarly (r.j)j = yj ......................... (2)
(r.k)k = zk .......................... (3)
(1) + (2) + (3)
(r.i)i + (r.j)j + (r.k)k = xi + yj + zk
(r.i)i + (r.j)j + (r.k)k =
2. By Vector method, prove that the angle in a semi circle is a right angle.
Proof: Let 0 be the centre and
AB be the diameter of the semi circle.
Let P be any point on the semi circle
Hence, the angle in a semi circle is a right angle.
Cosine Rule
3. In any ∆ ABC, if = BC, = CA and = AB then by vector method prove that
Proof : Given
a2 = b2 + c2 + 2bc cos(180° - A) [ (CA, AB) = 180° - A]
a2 = b2 + c2 - 2bc cos A
2bc cos A = b2 + c2 - a2
Projection Rule
4. In any triangle ABC, if a = BC, b = CA and c = AB then by vector method prove that a = b cos C + c cos B
Proof : Given
Similarly we can prove b = c cos A + a cos B
(Take = - - in place of (1) and dot with left side part)
c = a cos B + b cos A
(Take = - - in place of (1) and dot with left side part)
5. By vector method, prove that the altitudes of a triangle are concurrent
Proof: Altitude: Means the line segment drawn from a vertex to its line of opposite side.
Hence the altitudes of a triangle are concurrent.
6. By vector method, prove that the perpendicular bisectors of the sides of a triangle are concurrent.
Proof:
Hence the perpendicular bisectors of the sides of a triangle are concurrent.
7. By vector method prove that cos (A + B) = cos A cos B - sin A sin B. Where A and B are acute.
Proof: Let P (cos A, sin A) , Q (cos B, sin B) be any two points
from (1) and (2)
cos (A + B) = cosA cosB - sinA sinB
8. By vector method prove that cos (A - B) = cos A cos B + sin A sinB. Where A and B are acute.
Proof: Let P(cos A,sin A)
Q (cos B, sin B) be any two points