Final answer:
It is false that a matrix of 3 x 4 can be added to a matrix of 3 x 7; matrix addition requires identical dimensions. Additionally, vector addition properties include commutativity and the dependency of the resultant vector's magnitude and direction on the vectors being added.
Step-by-step explanation:
The statement that a matrix with dimensions of 3 x 4 can be added to a matrix with dimensions of 3 x 7 is false. In matrix addition, two matrices can be added together only if they have the same dimensions. That is, each matrix must have the same number of rows and the same number of columns.
For example, a matrix A with dimensions 3 x 4 can be added to another matrix B of dimensions 3 x 4, but not to a matrix C of dimensions 3 x 7 since the number of columns is different.
Regarding vector addition:
- The addition of any number of vectors is commutative, A + B = B + A, similar to ordinary numbers.
- The magnitude of the resultant vector from adding vectors does not necessarily increase with more vectors; it depends on their directions.
- True – The direction of the resultant vector depends on both the magnitude and direction of the added vectors.
- False – If only the angles of two vectors are known, we cannot find the angle of their resultant vector without additional information.
- True – Pythagorean theorem can be used to calculate the length of the resultant vector when two vectors are at right angles to each other.
- A vector can indeed form the shape of a right angle triangle with its x and y components.
- You cannot add a scalar to a vector because they are of different mathematical natures.
- Two vectors with different magnitudes can add to zero if they are in opposite directions and of equal magnitude.