Product Of Two Vectors

Step 1: Identity: ( a × b)² + ( a · b)² = a² b²
Step 2: a²(2)² = 676 a² = 169
Step 3: | a| = 13

Step 1: Required direction is a × b.
Step 2: a × b = 2( i+ j)
Step 3: Unit vector: ( i+ j)/(\sqrt2)

According to question and
Then ,

and
Required unit vector

Step 1: Analyze the given vector relations.
Given that and , we know that the cross product of two parallel vectors is zero. Hence, or .
Thus, the correct answer is (2).

Step 1: Find the cross product.
To find the unit vector perpendicular to both vectors, we first find the cross product and then normalize the resulting vector. The correct unit vector is .
Thus, the correct answer is (3).

Step 1: Use the dot product properties.
From the given conditions, we know that , so . Taking the dot product of both sides with themselves and simplifying gives us .
Step 2: Conclusion.
The value of is . Final Answer:

For three unit vectors equally inclined with mutual dot product ,

Let v=(x,y,z). Solve the linear system from dot products to get (x,y,z)=(1,2,3).

Let a=(1,2,1). a·(1,1,-3)=1+2-3=0 a·(1,3,-2)=1+6-2=5 a·(2,1,4)=2+2+4=8 Hence, the vector is i+2 j+ k.

Step 1: Understanding the Concept
The expression involves the Vector Triple Product. The formula for a triple product is .

Step 2: Key Formula or Approach
For the first term: . Since , then .

Step 3: Detailed Calculation
1. Magnitude of the first term: . 2. Similarly, for the second term: . 3. Similarly, for the third term: . 4. Summing them up: 5. Given , so . 6. Result: .

Step 4: Final Answer
The value of the expression is 18.