Question

Определить при каком А векторы А(1,3а,2) и вектор Б(2,3а,-3)будут взаимно перпендикулярны

Answer 1

Решение
Векторы перпендикулярны, если их скалярное произведение равно нулю:
A*Б = 1 * 2 + 3a*3a + 2*(-3) = 0
2 + 9a² - 6 = 0
9a² = 4
a² = 4/9
a1 = - 2/3
a2 = 2/3

Answer 2

Determining Perpendicular Vectors

To determine when the vectors A(1,3a,2) and B(2,3a,-3) are mutually perpendicular, we can use the fact that two vectors are perpendicular if and only if their dot product is zero.

The dot product of two vectors A and B is given by the formula: A · B = A_x * B_x + A_y * B_y + A_z * B_z, where A_x, A_y, A_z are the components of vector A, and B_x, B_y, B_z are the components of vector B.

So, for vectors A and B to be perpendicular, their dot product should be zero.

Let's calculate the dot product of vectors A and B and solve for the value of 'a' that makes them perpendicular.

The dot product of A and B is: A · B = 1 * 2 + 3a * 3a + 2 * (-3) = 2 + 9a^2 - 6

For A · B to be zero, the equation 2 + 9a^2 - 6 = 0 must hold true.

Solving for 'a': 9a^2 - 4 = 0 a^2 = 4/9 a = ±2/3

Therefore, the vectors A(1,3a,2) and B(2,3a,-3) will be mutually perpendicular when a = ±2/3.