Subtraction of vectors

4 Subtraction of vectors

Subtraction of one vector from another is performed by adding the corresponding negative vector. That is, if we seek $\underset{̲}{a} - \underset{̲}{b}$ we form $\underset{̲}{a} + (- \underset{̲}{b})$ . This is shown geometrically in Figure 18. Note that in the right-hand diagram the arrow on $\underset{̲}{b}$ has been reversed to give $- \underset{̲}{b}$ .

Figure 18 :

{ Subtraction of a vector is performed by adding a negative vector}

Exercises

Vectors $\underset{̲}{p}$ and $\underset{̲}{q}$ represent two perpendicular sides of a square. Find vector expressions which represent the diagonals of the square.
In the rectangle $A B C D$ , side $A B$ is represented by the vector $\underset{̲}{p}$ and side $B C$ is represented by the vector $\underset{̲}{q}$ . State the physical significance of the vectors $\underset{̲}{p} - \underset{̲}{q}$ and $\underset{̲}{p} + \underset{̲}{q}$ .
An object is positioned at the origin of a set of axes. Two forces act upon it. The first has magnitude 9 N and acts in the direction of the positive $y$ axis. The second has magnitude 4 N and acts in the direction of the negative $x$ axis. Calculate the magnitude and direction of the resultant force.
An object moves in the $x y$ plane with a velocity of $15 m s^{- 1}$ in a direction $4 8^{\circ}$ above the positive $x$ axis. Resolve this velocity into two components, one along the $x$ axis and one along the $y$ axis.

$\underset{̲}{p} + \underset{̲}{q}$ , $\underset{̲}{q} - \underset{̲}{p}$ . Acceptable answers are also $- (\underset{̲}{p} + \underset{̲}{q})$ , $\underset{̲}{p} - \underset{̲}{q}$ .
$\underset{̲}{p} + \underset{̲}{q}$ is the diagonal $A C$ , $\underset{̲}{p} - \underset{̲}{q}$ is the diagonal $D B$ .
Magnitude $\sqrt{97}$ , at an angle $6 6^{\circ}$ above the negative $x$ axis.
$10.04 m s^{- 1}$ along the $x$ axis, and $11.15 m s^{- 1}$ along the $y$ axis.

4 Subtraction of vectors

Exercises

Answer