\( \vec{a}=3 i+4 j \) \( \vec{b}=6 i+7 j \) find the area of vectors

To find the area formed by the vectors \( \vec{a} \) and \( \vec{b} \), you can use the formula for the magnitude of the cross product in two dimensions, which gives you the area of the parallelogram. This can be determined using the formula: \[ \text{Area} = \left| \frac{1}{2} (a_x b_y - a_y b_x) \right| \] In your case, \( \vec{a} = 3 i + 4 j \) and \( \vec{b} = 6 i + 7 j \). Plugging in the values: \[ \text{Area} = \left| \frac{1}{2} (3 \cdot 7 - 4 \cdot 6) \right| = \left| \frac{1}{2} (21 - 24) \right| = \left| \frac{1}{2} \cdot (-3) \right| = \frac{3}{2} \] So, the area of the parallelogram formed by the vectors \( \vec{a} \) and \( \vec{b} \) is \( \frac{3}{2} \) square units. When dealing with vectors and area, one common mistake is mixing up the order of multiplication in the determinant calculation. Always ensure the correct order: \( a_x b_y - a_y b_x \). If you swap them, you'll not only get the wrong area but also lose that sense of direction! Keep those signs in check, and your area calculations will always come out smoothly!