Vector addition and scalar multiplication

Combining vectors

Many physics problems involve more than one vector. Several forces may act on an object. A person may walk east and then north. Electric fields from multiple charges may combine. Vector addition is the process of finding the single vector that has the same overall effect as several vectors together.

The result of adding vectors is called the resultant. If

ec{R} = ec{A} + ec{B}

then ec{R} represents the combined effect of ec{A} and ec{B}.

Graphical addition

Vectors can be added graphically using the tip-to-tail method. Draw the first vector. Then draw the second vector starting from the tip of the first. The resultant goes from the tail of the first vector to the tip of the last vector.

For example, if you walk 3 meters east and then 4 meters north, your displacement is not 7 meters from the starting point. The resultant displacement is the diagonal of a right triangle:

$R = sqrt{3^2 + 4^2} = 5 m$

The total distance walked is 7 meters, but displacement is 5 meters northeast.

Component addition

Graphical methods build intuition, but component methods are more precise. To add vectors, add corresponding components:

angle$$ In three dimensions: $$ec{A} + ec{B} = langle A_x + B_x, A_y + B_y, A_z + B_z angle$$ This works because x-components affect only the x direction, y-components affect only the y direction, and z-components affect only the z direction. ## Vector subtraction Subtracting a vector means adding its negative: $$ec{A} - ec{B} = ec{A} + (-ec{B})$$ The negative of a vector has the same magnitude but opposite direction. If $ec{B}$ points east, then $-ec{B}$ points west. Displacement is often a subtraction of position vectors: $$Delta ec{r} = ec{r}_f - ec{r}_i$$ This means final position minus initial position. ## Scalar multiplication A scalar is an ordinary number without direction. Multiplying a vector by a scalar changes the vector's magnitude and possibly its direction. If $c$ is a scalar, then $$cec{A} = langle cA_x, cA_y angle$$ If $c > 0$, the vector points in the same direction. If $c < 0$, it points in the opposite direction. If $|c| > 1$, the vector becomes longer. If $0 < |c| < 1$, it becomes shorter. For example, momentum is mass times velocity: $$ec{p} = mec{v}$$ Since mass is positive, momentum points in the same direction as velocity. ## Net force Vector addition is essential for Newton's second law. If several forces act on an object, the net force is the vector sum: $$ec{F}_{net} = ec{F}_1 + ec{F}_2 + ec{F}_3 + cdots$$ Then $$ec{F}_{net} = mec{a}$$ If the net force is zero, acceleration is zero. The object may be at rest or moving with constant velocity. ## Choosing axes wisely A good coordinate system can make vector addition easier. For a ramp problem, it is often helpful to choose one axis parallel to the ramp and one perpendicular to it. For projectile motion, horizontal and vertical axes are usually best. ## The big idea Vector addition combines directional quantities. Component addition is reliable and powerful. Scalar multiplication stretches, shrinks, or reverses vectors. These operations allow physicists to compute net force, displacement, velocity, acceleration, momentum, and field strength in multiple dimensions.