Let us consider two vectors `vecA` and `vecB` that lie in a plane as shown in Fig. (a). The lengths of the line segments representing these vectors are proportional to the magnitude of the vectors.
To find the sum `vecA + vecB`, we place vector `vecB` so that its tail is at the head of the vector `vecA`, as in Fig. (b). Then, we join the tail of `vecA` to the head of `vecB`. This line OQ represents a vector `vecR`, that is, the sum of the vectors `vecA` and `vecB`. Since, in this procedure of vector addition, vectors are arranged head to tail, this graphical method is called the head-to-tail method.
The two vectors and their resultant form three sides of a triangle, so this method is also known as `text(triangle method of vector addition)`.
• If we find the resultant of `vecB + vecA` as in Fig. (c), the same vector `vecR` is obtained. Thus, vector addition is commutative.
`vecA + vecB = vecB + vecA`
• The addition of vectors also obeys the associative law as illustrated in Fig. (d). The result of adding vectors `vecA` and `vecB` first and then adding vector `vecC` is the same as the result of adding `vecB` and `vecC` first and then adding vector `vecA`.
`(vecA + vecB) + vecC = vecA + (vecB + vecC)`
`text(Null Vector or Zero Vector)`
Consider two vectors `vecA` and `–vecA`. Their sum is `vecA + (–vecA)`. Since the magnitudes of the two vectors are the same, but the directions are opposite, the resultant vector has zero magnitude and is represented by `vec0` called a `text(null vector)` or a `text(zero vector)`.
`vecA – vecA = vec0`
`|vec0|= 0`
Since the magnitude of a null vector is zero, its direction cannot be specified.
The null vector also results when we multiply a vector A by the number zero. The main properties of `vec0` are -
`vecA + vec0 = vecA`
`lamda vec0 = vec0`
`0 vecA = vec0`
`text(Subtraction of Vectors)`
Subtraction of vectors can be defined in terms of addition of vectors. We define the difference of two vectors `vecA` and `vecB` as the sum of two vectors `vecA` and `–vecB`.
`vecA – vecB = vecA + (–vecB)`
It is shown in Fig (e,f). The vector `–vecB` is added to vector `vecA` to get `vecR_2 = (vecA – vecB)`.
Let us consider two vectors `vecA` and `vecB` that lie in a plane as shown in Fig. (a). The lengths of the line segments representing these vectors are proportional to the magnitude of the vectors.
To find the sum `vecA + vecB`, we place vector `vecB` so that its tail is at the head of the vector `vecA`, as in Fig. (b). Then, we join the tail of `vecA` to the head of `vecB`. This line OQ represents a vector `vecR`, that is, the sum of the vectors `vecA` and `vecB`. Since, in this procedure of vector addition, vectors are arranged head to tail, this graphical method is called the head-to-tail method.
The two vectors and their resultant form three sides of a triangle, so this method is also known as `text(triangle method of vector addition)`.
• If we find the resultant of `vecB + vecA` as in Fig. (c), the same vector `vecR` is obtained. Thus, vector addition is commutative.
`vecA + vecB = vecB + vecA`
• The addition of vectors also obeys the associative law as illustrated in Fig. (d). The result of adding vectors `vecA` and `vecB` first and then adding vector `vecC` is the same as the result of adding `vecB` and `vecC` first and then adding vector `vecA`.
`(vecA + vecB) + vecC = vecA + (vecB + vecC)`
`text(Null Vector or Zero Vector)`
Consider two vectors `vecA` and `–vecA`. Their sum is `vecA + (–vecA)`. Since the magnitudes of the two vectors are the same, but the directions are opposite, the resultant vector has zero magnitude and is represented by `vec0` called a `text(null vector)` or a `text(zero vector)`.
`vecA – vecA = vec0`
`|vec0|= 0`
Since the magnitude of a null vector is zero, its direction cannot be specified.
The null vector also results when we multiply a vector A by the number zero. The main properties of `vec0` are -
`vecA + vec0 = vecA`
`lamda vec0 = vec0`
`0 vecA = vec0`
`text(Subtraction of Vectors)`
Subtraction of vectors can be defined in terms of addition of vectors. We define the difference of two vectors `vecA` and `vecB` as the sum of two vectors `vecA` and `–vecB`.
`vecA – vecB = vecA + (–vecB)`
It is shown in Fig (e,f). The vector `–vecB` is added to vector `vecA` to get `vecR_2 = (vecA – vecB)`.