Vectors Flashcards
What are vectors
They are quantities that have both magnitude and direction
How are vectors represented
👉🏾*with a arrow whose length signifies magnitude and arrowhead signify direction
Also written down in the form of i, j, and k which stands for their directions in the the x, y, and z axes
👉🏾It should be noted that there’s no vector like 3k, but 0i+0j+3k
*Also, negative signs means the vector is moving in opposite of direction
How do we rep vectors that doesn’t runs on the standard axes
*The lines that rep such would be at an angle to all the axes
*The magnitude of the vector on the axes would be such that:
*The axis touching the vector would be the magnitude of the vector multiplied by the cos of the angle between them
*And that of the axis with which it doesn’t make the angle would be the magnitude times the sine of the angle
Addition of vectors
*To sum strings of vectors, we assume a CW or ACW direction and sum all the vectors such that the vectors not going in the assumed direction take negative sign
*The i’s and j’s and k’s are collected together and summed together
*To get the resultant of such vector, we equate such sum above to zero.
Resolution of vectors
*Resultant of two vectors is a vector come from their origin
*and it has the same effect as the two
*It is usually simplified for using the parallelogram law of vector
*and solved for using the sine or cosine rules, depending on the givens
Multiplication of vectors
We have three types of vector multiplication
*Multiplying a vector by a scalar
*Multiplying vector with vector to yield scalar (dot product)
*Multiplying two vectors to yield a vector (cross product)
Vector quantities
*Any thing field is vector
*Anything flux is scalar, except the magnetic flux density
Multiplying a vector and a scalar
*It yeilds a vector in the same direction…
*whose magnitude is the scalar multiple of the vector
Dot product
*It is a multiplication of two vectors
*Yields a scalar quantity
👉🏾Also gotten as the products of f the two vectors and the cos of angle between them
*A.B is the same as B.A
Cross product
*Happens between vectors only
*Yeilds a vector quantity
*Gotten by finding the determinant of a matrix such that:
i. First column is i, j,k
Ii. Second column is the coefficients of i, j, and k in the first vector
iii. Third column is the coefficients of i, j, and k in the second vector
*Also calculated by doing expansion and solution; while noting that:
i. ixj=k; jxi=-k
Ii.jxk=i; kxj=-i
Iii.kxi=j;ixk=-j
Iv. ixi=jxj=kxk=0
AxB≠BxA (not dissociative).
👉🏾The magnitude of the resulting vector is such that:. It is the product of the magnitudes of the two vectors and the sine of the angle between them
*This is usually used to find the angle between them
What’s a Null vector
It is a vector whose magnitude is 0
