`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} "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`