Vectors

We are given the equation: Squaring both sides of the equation to eliminate the magnitudes: Expanding both sides: Simplifying: Since the dot product , this means that the vectors and are perpendicular. Thus, the correct answer is:

The projection of vector onto vector is given by the formula: First, calculate the dot product : Next, calculate the magnitude squared of : Now, use the formula for the projection: The magnitude of the projection is: Thus, the magnitude of the projection is:

We will use the distributive property of the cross product: Now substitute these into the original expression: Rearranging terms: Now, using the property that , we have: Thus, the terms cancel each other out: So, the entire expression equals zero: Thus, the correct answer is:

The scalar triple product of three vectors is given by: It represents the volume of the parallelepiped formed by the vectors . For the unit vectors , the cross product results in (from the right-hand rule and the cyclic property of unit vectors): Now, taking the dot product of with : Thus, the value of the scalar triple product is:

To find the unit vector in the direction of the vector , we use the formula: First, we calculate the magnitude of : Now, the unit vector is: Thus, the unit vector is: Thus, the correct answer is:

Two vectors are perpendicular if their dot product is zero. So, we calculate the dot product and set it equal to zero. The dot product is given by: Using the distributive property of the dot product: Simplifying the dot products: For the vectors to be perpendicular, the dot product must be zero: Solving for : Thus, the correct answer is:

The angle between two vectors and is given by the formula: First, calculate the dot product : Next, calculate the magnitudes of and : Now, use the formula for the cosine of the angle: Since , the angle is: Thus, the correct answer is:

The dot product of two vectors and is given by: Here, we have: Now, compute the dot product: Thus, the correct answer is:

The magnitude of a vector is given by: For the vector , we have , , and . So, the magnitude is: Thus, the correct answer is:

The dot product of two unit vectors and is defined as: where is the angle between the two vectors, and for unit vectors. For , the angle is 0 degrees (since they are the same vector), and . Therefore: Thus, the correct answer is:

The cross product of two unit vectors follows the right-hand rule and is based on the cyclic relationship of the unit vectors . The cyclic cross product rules are: For , we follow the right-hand rule and find: Thus, the correct answer is:

First, check if the planes are parallel. The normal vectors are: Since , the planes are parallel. Rewrite the second plane to match the first plane’s normal vector by dividing by 3: Distance between two parallel planes is given by: Rewrite planes as: Calculate:
Final answer:

The equation of a plane with intercepts on the axes respectively is: Substitute : Multiply both sides by 12 to clear denominators:
Final answer:

Direction ratios of the first line: Direction ratios of the second line: For the lines to be mutually perpendicular, their direction vectors must satisfy:
Final answer:

The angle between two vectors and is given by: Calculate the dot product: Calculate magnitudes: Therefore, Hence,