Engr Help

Vector Examples -

Ex. 1 Find the component of the vector, F₁ = 3i + 4j in the direction of the

vector, F₂ = 5i + 12j where i and j are unit vectors in the x and y

directions.

Solution: The magnitude of the vector F₂ is 13 (square root of the sum of the

squares of each component). So the unit vector in the direction of F₂ is

(5/13)i + (12/13)j . Recall from vector calculus the component of a vector, A,

in the direction of vector, B, is the dot product, A ● B / | B |

F₁ ● F₂/13 = ( 3i + 4j ) ● [(5/13)i + (12/13)j] = (15/13) + (48/13) = 63/13

So the magnitude of the resulting vector is 63/13.

Ex. 2 Find the vector of F₁ = 3i + 4j in the direction of the vector, F₂ where

F₂ = 5i + 12j and where i and j are unit vectors in the x and y directions.

Solution:

From the first example the magnitude of the resulting vector in the direction of

vector, F₂ is 63/13 and the unit vector in the direction of F₂ is (5/13)i + (12/13)j .

The resulting vector, F₃, in the direction of F₂ is just its magnitude times the unit vector

in the direction of F₂ .

So F₃ = (63/13) [(5/13)i + (12/13)j] = (315/13)i + 756/13)j

Note the same strategy applies to vectors in three dimensions with cartesian unit vectors

i, j, and k.

Find the sum of vectors A = 3i + 4j + 5k and B = 12i - 16j + 22k

Strategy: Add components.

A + B = (3i + 4j + 5k) + (12i - 16j + 22k) = 15i - 12j + 27k