Properties Of Vectors

We are asked to find the direction cosines of a vector that is perpendicular to both and .
Step 1: Find the cross product of and To find the direction cosines of a vector perpendicular to both and , we first compute the cross product . Given: The cross product formula is: Using the determinant formula for the cross product: Thus, the vector perpendicular to both and is . Step 2: Find the magnitude of The magnitude of the vector is: Step 3: Find the direction cosines The direction cosines of a vector are given by the components of the unit vector in the direction of . The unit vector is: Thus, the direction cosines are: Hence, the direction cosines of the vector perpendicular to and are .

We are given that the vectors and satisfy the equation: From this, we can express as: Next, we are given that and are unit vectors, so: Also, we are given that . Step 1: Find the magnitude of The magnitude of is: Now, square both sides: Since and , we get: Step 2: Find the angle between and We need to find the angle between and . Using the formula for the dot product: Substitute the known values: Now, substitute into the dot product: Equating the two expressions for : Thus, the angle is: Thus, the angle between and is .

To find the angle between two vectors and , we use the formula: Where:
- is the angle between the vectors
- is the dot product of the vectors
- and are the magnitudes of the vectors.
Step 1: Compute the dot product Given vectors: The dot product is: Step 2: Compute the magnitudes of and The magnitude of vector is: The magnitude of vector is: Step 3: Calculate the angle Now, we can find : Thus, . By calculating the inverse cosine, we get: Thus, the angle between the vectors is .