7/21/2019 Lecture 2 - Dot and Cross

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Vector Analysis – Lecture 2Engr. Michael M. Salvahan, ECE, ECT, CCAI

7/21/2019 Lecture 2 - Dot and Cross

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II. Dot and Cross roduct

Dot roduct Also called the scalar productGiven two vectors A and ! , dot product isdenoted by A"! , defined as the product ofmagnitudes of A and B and the cosine ofthe angle θ between them.

A"! # AB cos θ, 0 θ

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II. Dot and Cross roduct

The following laws are valid:$. A"! !"A !ommutative "aw2. A"%!&C' A"! # A"C $istributive "aw

%. m&A"! ' &m A' "! A(&m! ' where m is a scalar (. i"i )") *"* ) and i") )"* *"i 0*. +fA A ) i # A ) # A %* and ! B ) i # B ) # B %*,

then, A"! # A)B

) # A B # A

%B

%A"A # A 2 # A ) # A # A %

!"! # B B ) # B # B %

-. +f A"! 0 and A and ! are not null vectors,then A and ! are perpendicular.

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II. Dot and Cross roduct

Cross roduct Also called the vector productGiven two vectors A and ! , cross productis denoted by C A ! , defined as theproduct of magnitudes of A and ! and thesine of the angle θ between them.

A"! # AB sin θ u , 0 θ/here u is a unit vector indicating thedirection of A ! . +f A ! , or if A is parallel to! , then sin θ 0 and we define A ! 0.

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II. Dot and Cross roduct

The following laws are valid:$. A ! ! A2. A &! #C' A ! # A C%. m&A ! ' &m A' ! A &m! ' & A ! 'm(. i i ) ) * * 0, i ) *, ) * i, * i )

*. +fA A ) i # A ) # A %* and ! B ) i # B ) # B %*, then, A !

-. The magnitude of A ! is the same as the area of aparallelogram with sides A and ! .

1. +fA ! 0, and A and ! are not null vectors, then A and !are parallel.

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II. Dot and Cross roduct

Sa+ le ro-le+s). 2valuate each of the following.a. ) " %2i – /) & *'-. %2i – )' " %/i & *'

. 3ind the angle between A i # 4 5 6

and ! -i 5 %4 # 6Ans0ers

/. a. – /, -. 1

(. # 345

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II. Dot and Cross roduct

Sa+ le ro-le+s%. $etermine the value of a so that A i #

a4 # 6 and B 7i 5 4 5 6 are

perpendicular.

7. 8how that the vectors A %i 5 4 # 6, B i

5 %4 # *6, ! i # 4 5 76 form a righttriangle.Ans0ers

/. a # /

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II. Dot and Cross roduct

Sa+ le ro-le+s*. 3ind the angles which the vector A %i 5

-4 # 6 ma6es with the coordinate a es.

-. 3ind the pro4ection of the vector A i 5 4# 6 on the vector B 7i 5 74 # 16.1. 3ind the pro4ection of the vector B i # *4

# %6 on the vector A i # %4 # -6.

Ans0ers6. 7 # 1(.15, 8 # $(45, 9 # 3/.(5

1. $4:4

3. 6

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II. Dot and Cross roduct

Sa+ le ro-le+s9. 2valuate each of the following.a. & 4' &%6' c. 4 i 5 %6

b. &%i' & 6' d. i 6

. +f A i 5 %6 5 6 and B i # 74 5 6, find:a. A Bb. B Ac. &A # B' &A 5 B'

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II. Dot and Cross roduct

Sa+ le ro-le+s)0. +f A %i 5 4 # 6, B i # 4 5 6, and ! i 5

4 # 6 find:

a. &A B' !b. A &B !'Ans0ers

;. a. 1i, -. 1), c. <6*, d. –)4. a. $=i & /) & $$*, -. <$=i – /) – $$*,

c. <2=i – 1) – 22*$=. a. 2(i & 3) – 6*, -. $6i & $6) – $6*