Geometry And Vectors

Step 1: Calculate Box Product:

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Step 2: Dot Product with Basis Vectors:

Dot the main equation with , , :
Step 3: Sum Equations:

Summing the three equations:
Step 4: Final Answer:

The ratio is .

Step 1: Intersection Point P:

Line: . Plane: . Substitute: . P is .
Step 2: Plane Equation:

Normal . Plane: .
Step 3: Final Answer:

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Step 1: Vector Approach:

Use position vectors with respect to an origin. Midpoints: , .
Step 2: Detailed Explanation:

Sum . Substitute and :
Step 3: Final Answer:

The sum is .

Step 1: Check Coplanarity:

Let the points be P, Q, R, S with coordinates from coefficients: , , , . Vectors:


Step 2: Scalar Triple Product:

Since , vectors are coplanar.
Step 3: Final Answer:

The points are Coplanar.

Step 1: Understanding the Concept:

The normal vector of the required plane is perpendicular to the normal of the given plane and perpendicular to the vector joining the two points. We find this normal vector using the cross product.
Step 2: Key Formula or Approach:

Normal .
Step 3: Detailed Explanation:

Points and . Vector . Normal of given plane . Normal of required plane : . Direction ratios . Let's use for convenience. Equation of plane through : . Comparing with : . Compute expression: .
Step 4: Final Answer:

The value is -1.

Step 1: Understanding the Concept:

The problem asks for the shortest distance between two skew lines given in vector form. The shortest distance is measured along the line perpendicular to both direction vectors.
Step 2: Key Formula or Approach:

For lines and , the shortest distance is:
Step 3: Detailed Explanation:

Identify the components: Calculate : Calculate the cross product : Calculate the magnitude : Calculate the dot product : Calculate the distance:
Step 4: Final Answer:

The shortest distance is 8.

Step 1: Understanding the Concept:

We need to compute the cross product of two composite vectors. Instead of substituting the components immediately, it is more efficient to use the algebraic properties of the cross product (distributivity and anticommutativity) to simplify the expression first.
Step 2: Key Formula or Approach:

Properties of cross product: 1. 2. 3.
Step 3: Detailed Explanation:

Simplify the expression: Since and : (Using ) Now, calculate using the determinant method: Finally, multiply by -7:
Step 4: Final Answer:

The result is .

The expression in the image is ambiguous and appears to be ill-formed, likely due to a typesetting error. A standard interpretation like does not lead to any of the options.
Let's calculate :
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. This matches option (A), not (C).
Given the discrepancy, there is a high probability of a typo in the question's vectors or the intended expression. To justify the provided answer of 6, we must assume a different intended question.
Let's assume a typo in the vectors, for instance, if and .
Then .
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Let's consider another possibility. What if the expression was ?
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This expression would be . Not 6.
What if the expression was ? This gives . No.
Given the problem's flaws, we can only conclude that the question as written is incorrect. However, if forced to select an answer, and knowing that such errors are common, we acknowledge that the provided key is 6.

We are given the condition .
To work with this, we can square both sides of the equation.
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Using the property , we expand both sides.
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This simplifies to .
We are also given that . Let's substitute with .
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Subtract from both sides and add to both sides.
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This implies .
The dot product of two non-zero vectors is zero only if they are perpendicular to each other. So, the angle between and is .
We are told that vector is parallel to vector . This means has the same direction as .
Therefore, the angle between and is the same as the angle between and , which is .

The volume of a tetrahedron with coterminous edges given by vectors is , where is the scalar triple product.
Given , we have , which implies .
The scalar triple product can be calculated as the determinant of the matrix formed by the vectors.
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Expanding the determinant along the first row:
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Now we use the condition , so .
This gives two possibilities:
Case 1: .
Case 2: .
The values of p, which are the roots of the required quadratic equation, are 2 and -6.
To find the equation, we use the fact that an equation with roots is .
Sum of roots = .
Product of roots = .
The equation is , which simplifies to .

Let the position vectors of the vertices be .
Given: , , .
Let be the position vector of the centroid of face BCD. Given .
The formula for the centroid of triangle BCD is .
We can find the position vector of vertex D, , from this relation: .
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Now, find the position vector of the centroid of the tetrahedron, .
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We need to find the distance GD, which is the magnitude of the vector .
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Step 1: Understanding the Concept:

We determine the angle (or dot product) between the vectors using the given magnitudes and the difference vector, then use it to find the magnitude of the sum vector.
Step 2: Key Formula or Approach:

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Step 3: Detailed Explanation:

Given . Note that . This implies vectors and form a right-angled triangle with the difference vector as hypotenuse. Thus, they are perpendicular. . We need . Let this be .
Step 4: Final Answer:

The magnitude is .

Step 1: Understanding the Concept:

We need to find the equation of the plane and the parametric equation of the line, then solve for the intersection point.
Step 2: Key Formula or Approach:

Plane intercept form: . Line parametric form: .
Step 3: Detailed Explanation:

Plane Equation: Points are . Line Equation: Points and . Vector . Line: . Intersection: Substitute line into plane eq: Find Point: Vector: .
Step 4: Final Answer:

The point is .

Step 1: Understanding the Concept:

We first find the ratio in which P divides AB. If P divides AB in ratio , then the harmonic conjugate Q divides AB in ratio .
Step 2: Key Formula or Approach:

Section Formula: .
Step 3: Detailed Explanation:

Let P divide AB in ratio . Equating the i-components: P divides AB externally in ratio 1:3. Therefore, Q divides AB internally in ratio 1:3 ( ). Sum of scalar components:
Step 4: Final Answer:

The sum is 6.

Step 1: Understanding the Concept:

We solve the linear and quadratic equations to find the direction ratios of the two lines. The cosine of the angle is the dot product of the normalized direction vectors.
Step 2: Key Formula or Approach:

1. Substitution to find ratios. 2. .
Step 3: Detailed Explanation:

Given . Substitute into : Factorize: . Cases: 1. : . Ratios . 2. : . Ratios . Calculate : Vectors are and . Dot product: . Magnitudes: and . .
Step 4: Final Answer:

The value is .

Step 1: Understanding the Concept:

Using the Angle Bisector Theorem, point D divides BC in the ratio of the adjacent sides AB and AC. We calculate the lengths, find the coordinates of D, and then the length AD.
Step 2: Key Formula or Approach:

1. Distance formula: . 2. Section formula: where .
Step 3: Detailed Explanation:

Calculate lengths of AB and AC: . . Ratio . D divides BC internally in ratio . Coordinates of D: Length AD:
Step 4: Final Answer:

The length is .

Step 1: Find the value of 'a'.
The normal vectors to the two planes are and .
The angle between the planes is the angle between their normal vectors, given by .
Given , so .
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So, .
This gives .
Case 1: . . This is consistent with .
Case 2: . . The discriminant is , so no real solutions.
Thus, the only value is .
Step 2: Find the direction ratios of the line perpendicular to the third plane.
The third plane is . Substitute .
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The direction ratios of the normal to this plane are the coefficients of x, y, and z, which are (4, -2, 4).
A line perpendicular to this plane will have direction ratios parallel to the normal vector.
So the direction ratios of the line are proportional to (4, -2, 4).
We can simplify this by dividing by 2, which gives (2, -1, 2). This matches option (A).

Let the point be P(x, y, z).
The distance of a point P(x, y, z) from the XY-plane is the absolute value of its z-coordinate, which is .
The distance of a point P(x, y, z) from the Z-axis is the distance from P to the point (0, 0, z) on the Z-axis. This distance is .
The problem states that the distance from the XY-plane is twice the distance from the Z-axis.
So, we have the equation: .
To get the equation of the locus, we square both sides to eliminate the absolute value and the square root.
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Rearranging the terms to match the options, we get:
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Let the vertices of a cube of side length 'a' be placed at the origin (0,0,0) and along the positive axes.
Part 1: Find , the cosine of the angle between two body diagonals.
Let's take the diagonal from the origin (0,0,0) to (a,a,a), represented by vector .
And the diagonal from (a,0,0) to (0,a,a), represented by vector .
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Part 2: Find , the cosine of the angle between a body diagonal and a face diagonal.
Let's use the same body diagonal .
An intersecting face diagonal can be taken from the origin (0,0,0) to (a,a,0), represented by vector .
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Part 3: Calculate the required expression.
We need to find .
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Let be the angle between vectors and .
The expression we need to evaluate is .
By definition, the magnitude of the cross product is .
Substituting this into the expression, we get:
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We can find from the dot product formula: .
Given and .
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Using the Pythagorean identity, .
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Now, substitute the values back into our target expression:
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Now we must check which option gives the same numerical value. Let's evaluate option (C).
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Substitute and .
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Since the numerical result matches, option (C) is the correct answer.