Question

Векторы a(1;2;-m) и b(m; -3: 2) взаимно перпендикулярны. Найдите m

Answer 1

Векторы a(1;2;-m) и b(m; -3: 2) взаимно перпендикулярны. Найдите m

  Воспользуемся условием перпендикулярности двух векторов в пространстве в координатной форме.

Для перпендикулярности двух ненулевых векторов   и   необходимо и достаточно, чтобы их скалярное произведение равнялось нулю, то есть, чтобы выполнялось равенство:

1*m + 2*(-3) + (-m)*2 = 0,

-m – 6 = 0,

m = -6.

Answer 2

Given Information

We are given two vectors, a and b, and we know that they are mutually perpendicular. The components of vector a are (1, 2, -m), and the components of vector b are (m, ±3, 2).

Finding the Value of m

To find the value of m, we can use the fact that two vectors are perpendicular if their dot product is zero. The dot product of two vectors a and b is given by the formula:

a · b = a1 * b1 + a2 * b2 + a3 * b3

Since vector a and vector b are perpendicular, their dot product is zero:

(1 * m) + (2 * ±3) + (-m * 2) = 0

Simplifying the equation, we get:

m + 6 ± 2m = 0

Combining like terms, we have:

3m ± 6 = 0

Solving for m, we get two possible solutions:

For m + 6 = 0: m = -6

For -3m + 6 = 0: m = 2

Therefore, the value of m can be either -6 or 2.

Conclusion

The value of m can be either -6 or 2 for the vectors a(1, 2, -m) and b(m, ±3, 2) to be mutually perpendicular.