two equal vectors of magnitude 2A are inclime perpendicular to each other find the resultant and direction of vector.
2A sqrt(2) at an angle of 45o
You are given vectors A = 5.0i – 6.5j and B = -3.5i +7.0j. A third vector C lies in the xy-plane. Vector C is perpendicular to vector A, and the scalar product of C with B is 15.0. From this information, find the components of vector C.
To find the resultant of two equal vectors that are inclined perpendicular to each other, we can use the Pythagorean theorem and trigonometric ratios. Here's how you can solve it step by step:
Step 1: Draw a diagram representing the situation. Let's assume that one vector is pointing horizontally along the x-axis and the other vector is pointing vertically along the y-axis. Since the vectors are equal in magnitude and inclined perpendicular to each other, their components will also be equal and perpendicular.
Step 2: Determine the components of each vector. Since the vectors are equal and perpendicular, each vector's magnitude can be split equally between the x-axis (horizontal) and the y-axis (vertical). Therefore, the magnitude of each component is A.
Step 3: Apply the Pythagorean theorem. Since the components are perpendicular to each other, we can create a right-angled triangle where the magnitude of each component represents one side of the triangle. Using the Pythagorean theorem, we can find the magnitude of the resultant vector.
Magnitude of the resultant vector = √(A^2 + A^2) = √(2A^2) = √2A
Step 4: Determine the direction of the resultant vector. Since the components are perpendicular to each other, the resultant vector will form the hypotenuse of the right-angled triangle. To find the direction, we can use trigonometric ratios:
tanθ = Opposite/Adjacent = A/A = 1
Since tanθ = 1, we know that θ is equal to 45 degrees (common angle from trigonometry). Therefore, the direction of the resultant vector is 45 degrees relative to the positive x-axis.
So, the resultant vector has a magnitude of √2A and is inclined at an angle of 45 degrees with the positive x-axis.