Vector A has components Ax=4,Ay=3,Az=-5.
A, express A in terms of unit vector.
B, find the magnitude of A.
A. Expressing A in terms of unit vector notation:
If \( \hat{i} \), \( \hat{j} \), and \( \hat{k} \) are the unit vectors in the direction of the x, y, and z axes respectively, then the vector A can be expressed as:
\[ \textbf{A} = A_x\hat{i} + A_y\hat{j} + A_z\hat{k} \]
Given the components of the vector A (Ax = 4, Ay = 3, Az = -5), the vector A in unit vector notation is:
\[ \textbf{A} = 4\hat{i} + 3\hat{j} - 5\hat{k} \]
B. Finding the magnitude of A:
The magnitude (or length) of a vector A can be found using the Pythagorean theorem in three dimensions. Given the components of A (Ax, Ay, Az), the formula for the magnitude |A| is:
\[ |A| = \sqrt{A_x^2 + A_y^2 + A_z^2} \]
Using the given components of A:
\[ |A| = \sqrt{4^2 + 3^2 + (-5)^2} \]
\[ |A| = \sqrt{16 + 9 + 25} \]
\[ |A| = \sqrt{50} \]
So, the magnitude of A is:
\[ |A| = \sqrt{50} = 5\sqrt{2} \] (approximately 7.071)