Question

При каких значениях альфа,векторы а(12альфа-14альфа) b((4,3).А)перпендикулярно б) паралельно

Answer 1

Решение в прикрепленном файле.

Answer 2

Values of Alpha for Perpendicular and Parallel Vectors

To find the values of alpha for which the vectors a(12alpha-14alpha) and b(4,3) are perpendicular or parallel, we can use the properties of vector dot product.

Perpendicular Vectors: When two vectors are perpendicular, their dot product is zero.

The dot product of vectors a and b is given by: a • b = |a| * |b| * cos(theta)

Where: - a • b is the dot product of vectors a and b - |a| and |b| are the magnitudes of vectors a and b - theta is the angle between vectors a and b

For vectors to be perpendicular, the dot product a • b should be equal to 0.

Parallel Vectors: When two vectors are parallel, the angle between them is either 0 degrees or 180 degrees.

The dot product of parallel vectors is: a • b = |a| * |b| * cos(0) = |a| * |b| or a • b = |a| * |b| * cos(180) = -|a| * |b|

For vectors to be parallel, the dot product a • b should be equal to the product of the magnitudes of the vectors, or the negative of the product of the magnitudes of the vectors.

Calculating the Values of Alpha

Let's calculate the values of alpha for which the vectors a(12alpha-14alpha) and b(4,3) are perpendicular or parallel.

Perpendicular Vectors: For perpendicular vectors, the dot product a • b should be equal to 0.

a • b = (12alpha * 4) + (-14alpha * 3) = 0

Solving for alpha: 12alpha * 4 - 14alpha * 3 = 0 48alpha - 42alpha = 0 6alpha = 0 alpha = 0

So, the vectors a(12alpha-14alpha) and b(4,3) are perpendicular when alpha = 0.

Parallel Vectors: For parallel vectors, the dot product a • b should be equal to the product of the magnitudes of the vectors, or the negative of the product of the magnitudes of the vectors.

The magnitude of vector a is given by: |a| = sqrt((12alpha)^2 + (-14alpha)^2) = sqrt(144alpha^2 + 196alpha^2) = sqrt(340alpha^2) = 2sqrt(85) * |alpha|

The magnitude of vector b is given by: |b| = sqrt(4^2 + 3^2) = 5

So, the dot product a • b for parallel vectors is: a • b = 2sqrt(85) * |alpha| * 5

For the vectors to be parallel, the dot product a • b should be equal to the product of the magnitudes of the vectors, or the negative of the product of the magnitudes of the vectors.

2sqrt(85) * |alpha| * 5 = 5 * 2sqrt(85) * |alpha|

So, the vectors a(12alpha-14alpha) and b(4,3) are parallel for all values of alpha.

In summary: - The vectors are perpendicular when alpha = 0. - The vectors are parallel for all values of alpha.