Vector Algebra

is perpendicular to both and . Dot product of perpendicular vectors is . Hence .

Idea. The length of a vector is the 3-D version of Pythagoras: . Signs do not matter because we square.
Step.

Idea. Convert to components and use and cross terms .

Write components: and . Dot product multiply componentwise and add:

Unit vectors have length and are mutually perpendicular. Dot product of a vector with itself equals the square of its length:

Right–hand rule: . Swapping the order changes the sign:

Write components: and . Dot product . The correct value is . Since that number does not appear, the printed choices are inconsistent. If the question intended the magnitude only, (option (C)).

Here "square" means dotting the vector with itself (magnitude squared):

Magnitude . Unit vector vector (which simplifies to ).

Dot product: multiply components and add:

Compute componentwise: So the dot product equals .

Write components: , . Use determinant for cross product:

Magnitude of :

Use determinant: Compute: .

Step 1: Direction cosines are the cosines of the angles that a vector makes with the -, -, and -axes. Step 2: The -axis is the line along the -direction, so: - Angle with -axis = 90°, - Angle with -axis = 90°, - Angle with -axis = 0°, Therefore, direction cosines are . Final Answer:

First, find the cross product: Calculate: Evaluate minors: Now, take the dot product: Calculate:

Calculate : Magnitude:
Final answer:

First, find the magnitude of the vector: The direction cosines are given by:
Final answer:

The cross product . Therefore, we need to compute the dot product of with itself: Thus, the answer is 0.

We compute the dot product of two vectors: The dot product formula is: For the given vectors: Thus, the answer is 18.

We know that , and , as the cross product of a vector with itself is zero. Thus, the answer is .

In linear programming, the objective function is the function that we aim to either maximize or minimize based on the given constraints.
- Option (A) represents a constraint, not an objective function.
- Option (B) represents a constraint, not an objective function.
- Option (C) is a linear equation, which represents an objective function to either maximize or minimize.
Thus, the correct answer is: (C) .

The given vector is . The magnitude of a vector is given by: For the vector , we have , , and . Substituting these values into the formula for magnitude: Thus, the correct answer is option (B) .

The square of the vector is calculated as: Using the distributive property of the dot product: Since , , and (since the vectors are orthogonal): Thus, the correct answer is option (C) 25.

The dot product of two vectors and is given by: For the vectors and , we calculate: Thus, the correct answer is option (B) 24.

We are asked to evaluate the following expression involving dot products of unit vectors: Step 1: Recall the properties of dot products for unit vectors: - - - - , , and because the unit vectors are perpendicular to each other. Step 2: Substitute these values into the expression: Thus, the correct answer is: (C) 3.

The dot product of the vectors and is given by: Thus, the correct answer is option (B) 23.