● Multiplication of two vectors is also defined in two ways, namely, scalar (or dot) product where the result is a scalar, and vector (or cross) product where the result is a vector.

`color{blue} "Definition"` The scalar product of two nonzero vectors `veca` and `vecb` , denoted by `vec a ⋅ vecb` , is

defined as `color{blue}{veca ⋅ vecb = | vec a | | vecb | cosθ}`,

where, `θ` is the angle between `vec a` and `vec b`, `0 ≤ θ ≤ π` (Fig).



`\color{green} ✍️`If either `vec a = vec0` or `vecb = vec0`, then `θ` is not defined, and in this case,

we define `vec a ⋅ vec b = 0`

`color{red}{"Point To Consider :"}`

1. `vec a ⋅ vec b` is a real number.

2. Let a and b be two nonzero vectors, then `color{blue}{vec a ⋅ vec b = 0}` if and only if `veca` and `vec b` are `color{blue}{"perpendicular"}` to each other. i.e.

`veca * vec b = 0 => vec a bot vec b`

3. If `θ = 0,` then `veca ⋅ vec b = |vec a | | vecb |`

In particular, `veca ⋅ veca =| veca |^2` , as θ in this case is 0.

4. If `θ = π,` then `vec a ⋅ vecb = −| veca | |vec b |`

In particular, `veca ⋅ (−veca) = −| veca |^2` , as `θ` in this case is `π.`

5. For `hat i , hat j ` and `hat k`, we have

`hat i ⋅ hat i = hatj ⋅ hatj = hat k ⋅ hat k =1`,
` hati ⋅ hatj = hat j ⋅ hat k = hat k ⋅ hat i = 0`

6. The angle between two nonzero vectors `veca` and `vec b` is given by

`cos theta = ( vec a * vec b)/( | vec a | | vec b | )` or `color{blue}theta = cos^-1 ( (vec a * vec b )/(|vec a | | vec b | ) )}`

7. The scalar product is commutative. i.e.

`vec a * vec b = vec b * vec a`

`color {blue} "Property 1"` (Distributivity of scalar product over addition) Let `veca, vecb` and `vec c` be any three vectors, then

`vec a * ( vec b + vec c ) = vec a * vec b + vec a * vec c`

`color {blue} "Property 2"` Let `vec a` and `vec b` be any two vectors, and λ be any scalar. Then

`(λ vec a) ⋅ vec b = (λ vec a) ⋅ vec b = λ(vec a ⋅ vec b) = vec a ⋅ (λ vec b)`

`=>` If two vectors `vec a` and ` vec b` are given in component form as `a_1 hat i + a_2 hat j + a_3 hat k` and ` b_1 hat i + b_2 hat j + b_3 hat k` , then their scalar product is given as

`vec a ⋅ vec b = (a_1 hati + a_2 hatj + a_3 hat k)⋅(b_1 hat i + b_2 hat j + b_3 hat k)`

`= a_1 hat i ⋅ (b_1 hat i + b_2 hatj + b_3 hat k) + a_2 hatj ⋅ (b_1 hat i + b_2 hatj + b_3 hat k) + a_3 hat k ⋅ (b _1 hati + b_2 hatj + b_3 hat k)`

`= a_1 b_1 ( hat i ⋅ hat i) + a_1 b_2 (hat i ⋅ hatj) + a_1 b_3 (hat i ⋅ hat k) + a_2 b _1( hat j ⋅ hat i) + a_2 b_2 ( hat j ⋅ hatj) + a_2 b_3 ( hat j ⋅ hatk)`
`+ a_3 b_1 (hat k ⋅ hat i) + a b (hat k ⋅ hat j) + a_3 b_3 (hat k ⋅ hat k)` (Using the above Properties 1 and 2)

`= a_1b_1 + a_2b_2 + a_3b_3` (Using Observation 5)

Thus ` color{blue}{vec a ⋅ vec b = a_11b_1 + a_2b_2 + a_3b_3}`

`\color{green} ✍️` A vector `vec(AB)` makes an angle θ with a given directed line l (say), in the anticlockwise direction (Fig).

`=>` Then the projection of `vec(AB)` on l is a vector `vec p` (say) with magnitude `|vec ( AB)| cosθ`, and the direction of `vec p` being the same (or opposite) to that of the line l, depending upon whether cos θ is positive or negative.



The vector `vec p` is called the projection vector, and its magnitude `|vec p|` is simply called as the projection of the vector AB on the directed line l.

For example, in each of the following figures (Fig (i) to (iv)), projection vector of `vec (AB)` along the line l is vector `vec (AC)`.

`\color{green} {✍️ "Observations"} `

1. If `hat p` is the unit vector along a line l, then the projection of a vector `vec a` on the line l is given by `vec a ⋅ hat p` .

2 . Projection of a vector `vec a` on other vector `vec b` , is given by

`color{orange}{ vec a * ( vec b/ (| vec b | )) , "or" \ \ 1/ (| vec b |) ( vec a * vec b)}`

3. If `θ = 0,` then the projection vector of `vec (AB)` will be `vec(AB)` itself and if θ = π, then the projection vector of `vec (AB)` will be `vec (BA)`

4. If `theta = pi/2 ` or ` theta = (3 pi ) /2 ` , then the projection vector of `vec (AB)` will be zero vector.


`color {blue }"Remarks"` If `α, β` and `γ` are the direction angles of vector `vec a = a_1 hat i + a_2 hat j + a_3 hat k` , then its direction cosines may be given as

`cos alpha = ( vec a * hat i )/ ( | vec a | | hat i | ) = a_1/ (| vec a | ), cos beta = a_2 /( | vec a | ) `, and `cos gamma = a_3 / (| vec a | )`


`"Note :"` `|vec a | cosα, |vec a |cosβ` and `|vec a| cosγ` are respectively the projections of `vec a ` along OX, OY and OZ. i.e., the scalar components `a_1, a_2` and `a_3` of the vector `vec a`, are precisely the projections of `veca` along x-axis, y-axis and z-axis, respectively. Further, if `vec a` is a unit vector, then it may be expressed in terms of its direction cosines as `vec a = cosα hat i + cosβ hat j + cos γ hat k`