vector quantity
A parallelogram of vectors. Vectors can be added graphically using the parallelogram rule. According to the rule, the sum of vectors p and q is the vector r which is the diagonal of the parallelogram with sides p and q.
Any physical quantity that has both magnitude (size) and direction, such as velocity, acceleration, or force, as distinct from a scalar quantity such as speed, density, or mass, which has magnitude but no direction. A vector is represented either geometrically by an arrow whose length corresponds to its magnitude and points in an appropriate direction, or by two or three numbers representing the magnitude of its components. Vectors can be added graphically by constructing a parallelogram of vectors (such as the parallelogram of forces commonly employed in physics and engineering). This will give a resultant vector.
| The position vector of a point A(x,y) represents the move from the origin to A, that is a translation. A free vector has magnitude and direction but no fixed position in space. |
If two forces p and q are acting on a body at A, then the parallelogram of forces is drawn to determine the resultant force and direction r. p, q, and r are vectors. In technical writing, a vector is denoted by bold type, underlined, or overlined.
Equal vectors Two vectors are equal if they have the same size and the same direction. Equal vectors can be labelled with the same letter: |
| −a is the same size as a but points in the opposite direction. |
| a + b is the same as a followed by b. Vectors can only be added ‘nose to tail’, so the ‘nose’ (end point) of a is joined to the ‘tail’ (start point ) of b. The vectors a, b, and a + b look like this: |
| This is called a vector triangle. |
| For example, to subtract column vectors the numbers in the corresponding position are subtracted: |
Multiplication of vectors A vector can be multiplied by a scalar quantity. For example: |
| If p and q are scalars then qa + pb is a linear combination of a and b. For example, 2a + 3b is a linear combination of a and b where p = 2 and q = 3. |
Vector addition in practice Vectors can be used for practical purposes, for instance to find the velocity (speed in a particular direction) of a boat crossing a river. The boat is being rowed at 2.5 m per second, and the river is flowing from right to left at 1 m per second. The boat actually travels at an angle, instead of perpendicular to the river, and it also travels faster because the speed of the river increases its speed. To find the boat's velocity two vectors need to be added together. |
| The vector triangle for the boat's journey is: |
| √(12 + 2.52) = √7.25 = 2.7 m per second (to one decimal place) |
| tan θ = 1/2.5 = 0.4 θ = 21.8° (to one decimal place) |
| The resultant vector is the boat's velocity across the river, which is 2.7 m per second at 21.8° to the left of the perpendicular. |
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