MATH 127
Vector Analysis
♦ Vector addition: The sum of two vector →A and →B is a vector →C , which obtained by placing the initial point of →B on the final point of →A and the drawing a line from the initial point of →A to the final point of →B .
picture
♦ Scaler product: The product of a scaler say m times a vector
→A
is another vector
→B
, where
→B
has the direction as
→A
but the magnitude is changed that is
|→B|=m|→A|
• Properties:
1. Commutative law for addition: →A+→B=→B+→A
2. Associative law for addition: →A+(→B+→C)=(→A+→B)+→C
3. Commutative law for multiplication: m→A=→Am
4. Associative law for multiplication: (m+n)→A=m→A+n→A , where m and n are two different scalers.
5. Distributive law: m(→A+→B)=m→A+m→B
♦ Vector Multiplication: There are two types of multiplication.
1. Scaler or Dot product 2. Vector or Cross product
•Dot Product:
A geometric representation of dot product: The scaler or dot product of two vectors, →A and →B denoted by →A.→B , is defined as the product of the magnitude of vector times the cosine of the angle between them.
→A.→B=|→A| |→B|cosθ [θ=mnimum angle]
A cartesian representation of dot product: textrmI→A=A1ˆi+A2ˆj+A3ˆk and →B=B1ˆi+B2ˆj+B3ˆk then →A→B=A1B1+A2B2+A3B3
• Properties of Dot product:
1. →A.→B=→B.→A
2. →A.(→B+→C)=→A.→B+→A.→C
3. m(→A.→B)=(m→A.→B)=→A.(m→B)=(→A.→B)m
• Some Important results:
1. ˆi.ˆi=|ˆi||ˆi|cos0=1 ˆj.ˆj=1 ˆk.uvjk=1
2. ˆi.ˆj=|ˆi|.|ˆj|cos90=0 ˆj.ˆk=0 ˆk.ˆi=0
3. If we have →A.→B=0 means, neither the magnitude of →A nor →B is 0, then →A and →b must be perpendicular.
→A.→B=0and|→A|≠0 and |→B|≠0 then, θ=90∘
Example: What is the angle between of two vectors →A=2ˆi+2ˆj and →B=4ˆi−3ˆj ?
Solution: →A.→B=12−6=6|→A|=√22+22=2√2|→B|=√62+(−3)2=√45 ∴θ=cos−1(62√90) =71.56∘
♦ Cross product:
A geometric representation: The cross product of two vector →A and →B denoted by →A×→B is defined as the product of the magnitudes of the vectors times the sine of the angle between them with a unit vector ˆn .
→A×→B=|→A|.|→B|sinθ ˆn is the unit vector at right angles to both →A and →A
A cartesian representation:
If →A=A1ˆi+A2ˆj+A3ˆk→B=B1ˆi+B2ˆj+B3ˆk
Then →A×→B=|ˆiˆjˆkA1A2A3B1B2B3|=ˆi(A2B3−B2A3)+ˆj(A3B1−A1B3)+ˆk(A1B2−A2B1)
Example: If
→A=2ˆi+ˆj−ˆk and →B=−3ˆi+4ˆj+ˆk
find
→A×→B
Solution:
→A×→B=|ˆiˆjˆk21−1−341|=6ˆi+ˆj+11ˆk
• Properties of Cross product:
1. If →A×→B=0 then →A and →B will be parallel vector.
2. →A×→B≠0 provided that →A×→B is orthogonal to both →A and →B
3. If →u,→v and →w vectors and c is a number then
i) →u×→v=−(→v×→u)ii) →u×(→v+→w)=→u×→v+→u×→wiii) (c→u)×→v=uvu×(c→v)=c(→u×→v)iv) →u(→v×→w)=(→u×→v)→wv) →u(→v×→w)=|u1u2u3v1v2v3w1w2w3|
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