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Multiplication of Vectors by a Real Number

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Multiplication of Vectors by real numbers

Multiplying a vector A with a positive number λ gives a vector whose magnitude is changed by the factor λ but the direction is the same as that of A: |λA| = λ|A|     if λ > 0
For example, if A is multiplied by 2, the resultant vector 2A is in the same direction as A and has a magnitude twice of |A| as shown in Fig. (a) below.
Multiplying a vector A by a negative number −λ gives another vector whose direction is opposite to the direction of A and whose magnitude is λ times |A|.
Multiplying a given vector A by negative numbers, say –1 and –1.5, gives vectors as shown in Fig (b) below.
The factor λ by which a vector A is multiplied could be a scalar having its own physical dimension. Then, the dimension of λ A is the product of the dimensions of λ and A. For example, if we multiply a constant velocity vector by duration (of time), we get a displacement vector.

Dot product or scalar product:-
The dot product of two vectors A and B, is a scalar, which is equal to the product of the magnitudes of A and B and the Cosine of the smaller angle between them. If Θ is the smaller angle between A and B, then AB = AB
`bar A.bar B=ABcosθ`
(i) `hat i.hat i=hat j. hat j=hat k.hat k=1`
(ii) `hat i.hat j=hat j.hat k=hat k.hat i=0`
(iii) If `barA=A_x hat i +A_y hat j +A_z hat k and barB=B_x hat i +B_y hat j +B_z hat k`
then `barA .bar B=A_x B_x +A_y B_x +A_z B_z`

Properties of Scalar product:-
1. It obeys commutative law.
`barA .bar B=barB .bar A`

2. It obeys distributive law.
`barA .(bar B + bar C)= barA .bar B+ barA .bar C`

3. Scalar (Dot) product of two mutually perpendicular vectors is zero i.e.
`(barA .bar B) = AB cos 90^o=0`

4. Scalar (Dot) product will be maximum when θ = 0o i.e., vectors are parallel to each other.
`(barA .bar B)_(max)= |A||B|`

5. If `bar a ` and `bar b` are unit vectors then `|bar a|=|bar b|=1`and `bar a.bar b=1.1 cos θ= cos θ`

6. Dot product of unit vectors `hat i, hat j, hat k`
`hat i. hat i =hat j. hat j =hat k. hat k = 1
hat i. hat j =hat j. hat k =hat k. hat i = 0`

7. Square of a vector `bar a . bar a =|a||a| cos 0=a^2`

8. If the two vectors `bar A` and `bar B`, in terms of their rectangular components, are `barA=A_x hat i +A_y hat j +A_z hat k and barB=B_x hat i +B_y hat j +B_z hat k`

then `barA .bar B=(A_x hat i +A_y hat j +A_z hat k)(B_x hat i +B_y hat j +B_z hat k)`
        `barA .bar B=A_x B_x +A_y B_x +A_z B_z`


Vector product(Cross product):
The cross product of two vectors and, represented by x is a vector, which is equal to the product of the magnitudes of A and B and the sine of the smaller angle between them. If Θ is the smaller angle between A and B, then = AB Sin θ
where `hat n` is a unit vector perpendicular to the plane containing `bar A` and `bar B`.
(i) `hat i × hat i =hat j × hat j =hat k × hat k = 0`
(ii) `hat i × hat j =hat k,  hat j × hat k =hat i,    hat k × hat i =hat j`
     `hat j × hat i =`-`hat k`, `hat k × hat j =`-`hat i, hat i × hat k =`-`hat j`
(iii) If `barA=A_x hat i +A_y hat j +A_z hat k and barB=B_x hat i +B_y hat j +B_z hat k`
`bar A × bar B = (A_x B_z- A_z B_y) hat i +(A_z B_x- A_x B_z) hat j +(A_x B_y- A_y B_x) hat k `

Properties of Cross Product
(i) Cross product of two vectors is not commutative
`bar a × bar b ≠ bar b × bar a`
`bar a × bar b =-bar b × bar a`

(ii) Cross product is not assosiative
`bar a × (bar b × bar c) ≠ (bar a × bar b) × bar c`

(iii) Cross product obeys distributive law
`bar a × (bar b + bar c) = bar a × bar b + bar a × bar c`

(iv) If θ = 0 or `pi` it means the two vectors are collinear.
`bar a × bar b = bar 0`
and conversely, if `bar a × bar b = bar 0` then the vector `bar a` and `bar b` are parallel provided `bar a` and `bar b` are non-zero vectors.

(v) If  θ = 90o, and `hat n` is the unit vector perpendicular to both `bar a` and `bar b` 
`bar a × bar b=|a||b|  sin 90^o hat n =|a||b|  hat n`

(vi) The vector product of any vector with itself is `bar 0`
`bar a × bar a = bar 0`

(vii) If `bar a × bar b = bar 0`,then
`bar a=0 or bar b =0 or bar a|| bar b`

(viii) If `bar a ` and `bar b` are unit vectors, then `bar a × bar b=1.1 sin θ  hat n = sin θ  hat n`

(ix) Cross product of unit vectors `hat i, hat j and hat k`
`hat i × hat i =hat j × hat j =hat k × hat k = 0`

`hat i × hat j =hat k = - hat j × hat j`
`hat j × hat k =hat i = -hat k × hat j`
`hat k × hat i =hat j = -hat i × hat k`

(x) If the two vectors `bar A` and `bar B` in terms of their rectangular components are
`bar A = a_1 hat i + b_1 hat j + c_1 hat k`
`bar B = a_2 hat i + b_2 hat j + c_2 hat k`
`bar A × bar B  = (a_1 hat i + b_1 hat j + c_1 hat k)×( a_2 hat i + b_2 hat j + c_2 hat k`)
It can be found by the determinant method
i.e.,    `bar A × bar B =[(hat i,hat j,hat k),("a"_1 ,"b"_1,"c"_1),("a"_2,"b"_2,"c"_2)]`
`= hat i(b_1 c_2 - b_2 c_1) - hat j(a_1 c_2 - a_2 c_1) + hat k(a_1 b_2 - a_2 b_1) `

  • For motion in a plane, velocity is defined as:
    `bar v =(bar r_2 - bar r_1)/(t_2 - t_1)= ((x_2 hat i + y_2 hat j) - (x_1 hat i + y_1 hat j))/((t_2 - t_1)) = (x_2 - x_1)/(t_2 - t_1) hat i + (y_2 - y_1)/(t_2 - t_1) hat j = v_x hat i and v = sqrt(a_x^2 + a_y^2) `
  • For motion in a plane, acceleration is defined as:
    `bar a =(bar v_2 - bar v_1)/(t_2 - t_1)= ((v_(x_2) hat i + v_(y_2) hat j) - (v_(x_1) hat i + v_(y_1) hat j))/((t_2 - t_1)) = ((v_(x_2) - v_(x_1))/(t_2 - t_1)) hat i + ((v_(y_2) - v_(y_1))/(t_2 - t_1)) hat j and v = sqrt(a_x^2 + a_y^2) `
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