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Vector Addition and Subtraction Suppose a particle undergoes a displacement followed by a second displace￾ment . The final result is the same as if the particle had started at the same initial point and undergone a single displacement (Fig. 1.11a). We call displacement the vector sum, or resultant, of displacements and We express this rela￾tionship symbolically as (1.2) The boldface plus sign emphasizes that adding two vector quantities requires a geometrical process and is not the same operation as adding two scalar quantities such as In vector addition we usually place the tail of the second vector at the head, or tip, of the first vector (Fig. 1.11a). If we make the displacements and in reverse order, with first and sec￾ond, the result is the same (Fig. 1.11b). Thus (1.3) This shows that the order of terms in a vector sum doesn’t matter. In other words, vector addition obeys the commutative law. Figure 1.11c shows another way to represent the vector sum: If vectors and are both drawn with their tails at the same point, vector is the diagonal of a parallelogram constructed with and as two adjacent sides. CAUTION Magnitudes in vector addition It’s a common error to conclude that if

We will solve first for the x and y components of the resultant vector using the following formula: We will solve first for the x and y components of the resultant vector using the following formula: We will solve first for the x and y components of the resultant vector using the following formula: We will solve first for the x and y components of the resultant vector using the following formula: We will solve first for the x and y components of the resultant vector using the following formula: We will solve first for the x and y components of the resultant vector using the following formula: We will solve first for the x and y components of the resultant vector using the following formula:

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For many specific vector spaces, the vectors have received specific names, which are listed below. Historically, vectors were introduced in geometry and physics (typically in mechanics) before the formalization of the concept of vector space. Therefore, one talks often of vectors without specifying the vector space to which they belong. Specifically, in a Euclidean space, one consider spatial vectors, also called Euclidean vectors which are used to represent quantities that have both magnitude and direction, and may be added and scaled (that is multiplied by a real number) for forming a vector space.

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