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Chapter 1: Problem 126

Consider points \(A, B, C\) and \(D\) with position vectors \(7 \hat{\mathbf{i}}-4 \hat{\mathbf{j}}+7 \hat{\mathbf{k}}, \hat{\mathbf{i}}-6 \hat{\mathbf{j}}+10 \hat{\mathbf{k}},-\hat{\mathbf{i}}-3 \hat{\mathbf{j}}+4 \hat{\mathbf{k}}\) and \(5 \hat{\mathbf{i}}-\hat{\mathbf{j}}+5 \hat{\mathbf{k}}\) respectively. Then, \(A B C D\) is a [AIEEE 2003] (a) square (b) rhombus (c) rectangle (d) None of these

Short Answer

Expert verified
(d) None of these

Step by step solution

01

Calculate Vector Magnitudes

Define vectors for the sides: \( \overrightarrow{AB} = \overrightarrow{B} - \overrightarrow{A} = (-6\hat{\mathbf{i}} - 2\hat{\mathbf{j}} + 3\hat{\mathbf{k}}) \), \( \overrightarrow{BC} = \overrightarrow{C} - \overrightarrow{B} = (-2\hat{\mathbf{i}} + 3\hat{\mathbf{j}} - 6\hat{\mathbf{k}}) \), \( \overrightarrow{CD} = \overrightarrow{D} - \overrightarrow{C} = (6\hat{\mathbf{i}} + 2\hat{\mathbf{j}} + \hat{\mathbf{k}}) \), and \( \overrightarrow{DA} = \overrightarrow{A} - \overrightarrow{D} = (2\hat{\mathbf{i}} - 3\hat{\mathbf{j}} + 2\hat{\mathbf{k}}) \). Calculate magnitudes: \( \| \overrightarrow{AB} \| = \sqrt{6^2 + 2^2 + 3^2} = 7 \), \( \| \overrightarrow{BC} \| = \sqrt{2^2 + 3^2 + 6^2} = 7 \), \( \| \overrightarrow{CD} \| = \sqrt{6^2 + 2^2 + 1^2} = 7 \), and \( \| \overrightarrow{DA} \| = \sqrt{2^2 + 3^2 + 2^2} = 3 \sqrt{3} \). The magnitudes of \( \overrightarrow{AB}, \overrightarrow{BC}, \overrightarrow{CD}, \) are equal, but \( \overrightarrow{DA} \) is not.
02

Calculate Diagonal Magnitudes

Compute the diagonals: \( \overrightarrow{AC} = \overrightarrow{C} - \overrightarrow{A} = (-8\hat{\mathbf{i}} + \hat{\mathbf{j}} - 3\hat{\mathbf{k}}) \) and \( \overrightarrow{BD} = \overrightarrow{D} - \overrightarrow{B} = (4\hat{\mathbf{i}} + 5\hat{\mathbf{j}} - 5\hat{\mathbf{k}}) \). Diagonal magnitudes are \( \| \overrightarrow{AC} \| = \sqrt{8^2 + 1^2 + 3^2} = 6 \sqrt{2} \) and \( \| \overrightarrow{BD} \| = \sqrt{4^2 + 5^2 + 5^2} = \sqrt{66} \). These are not equal.
03

Analyze Given Options

The lengths \( \overrightarrow{AB}, \overrightarrow{BC}, \overrightarrow{CD} \) are equal to 7, but \( \overrightarrow{DA} \) is different. Any rectangle or square should have equal lengths of diagonals and respective sides should match certain patterns, which isn't the case here. Some side lengths are equal, but diagonals differ, thus it cannot be a square, rhombus, or rectangle.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Position Vectors
Position vectors are a cornerstone of vector geometry. They represent the position of a point in space relative to an origin, usually denoted by the letters \( \hat{i} \), \( \hat{j} \), and \( \hat{k} \).
  • \( \hat{i} \) represents the x-axis direction.
  • \( \hat{j} \) represents the y-axis direction.
  • \( \hat{k} \) represents the z-axis direction.
To find the position vector of a point, you start from the origin and move according to the components along these three axes. For example, the position vector of point \( A \) is \( 7 \hat{i} - 4 \hat{j} + 7 \hat{k} \), which means moving 7 units in the x-direction, -4 units in the y-direction, and 7 units in the z-direction. Every position vector provides a clear and precise way to describe the location of a point in three-dimensional space.
Vector Magnitude
The magnitude of a vector is essentially its length or size. You can think of it as the 'distance' the vector covers in space. Calculating the magnitude involves a simple formula derived from the Pythagorean theorem.For a vector \( \overrightarrow{v} = a \hat{i} + b \hat{j} + c \hat{k} \), the magnitude \( \| \overrightarrow{v} \| \) is given by:\[\| \overrightarrow{v} \| = \sqrt{a^2 + b^2 + c^2}\]This formula allows us to calculate the magnitudes of the various vectors connecting points \( A, B, C, \) and \( D \). For example, the magnitude of vector \( \overrightarrow{AB} \) is calculated as:\[\| \overrightarrow{AB} \| = \sqrt{(-6)^2 + (-2)^2 + 3^2} = 7\]Understanding vector magnitude is crucial because it forms the basis for comparing distances and lengths in vector geometry problems.
Diagonal Calculation
Diagonal calculations are important in vector geometry when analyzing shapes like quadrilaterals. Diagonals connect non-adjacent vertices of a polygon and provide critical information about the shape.For a quadrilateral \( ABCD \), there are two diagonals: \( \overrightarrow{AC} \) and \( \overrightarrow{BD} \). You find these by subtracting position vectors:
  • \( \overrightarrow{AC} = \overrightarrow{C} - \overrightarrow{A} \)
  • \( \overrightarrow{BD} = \overrightarrow{D} - \overrightarrow{B} \)
Then, calculate the magnitude of each diagonal using the same magnitude formula:\[\| \overrightarrow{AC} \| = \sqrt{(-8)^2 + 1^2 + (-3)^2} = 6 \sqrt{2}\]\[\| \overrightarrow{BD} \| = \sqrt{4^2 + 5^2 + (-5)^2} = \sqrt{66}\]The magnitudes of the diagonals tell us about the symmetry (or lack thereof) of the shape, which helps in determining if the quadrilateral is a square, rectangle, or another shape.
Quadrilaterals
Quadrilaterals are four-sided polygons, and understanding their properties is essential in vector geometry. The classification of a quadrilateral鈥攚hether it can be a square, rectangle, rhombus, or none of these鈥攄epends significantly on the lengths of its sides and diagonals.Key characteristics:
  • Sides: In a square, all sides are equal. For a rectangle, opposite sides are equal. A rhombus has all sides equal, but not necessarily the diagonals.
  • Diagonals: Squares and rectangles have equal diagonals, but a rhombus does not.
In our example, the lengths of sides \( \overrightarrow{AB}, \overrightarrow{BC}, \overrightarrow{CD} \) were equal, but \( \overrightarrow{DA} \) was not, ruling out a square or a rhombus. Additionally, unequal diagonals indicated that it wasn't a rectangle. Thus, the quadrilateral \( ABCD \) does not fit the standard categories and is likely none of the given options.

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Most popular questions from this chapter

The point having position vectors \(2 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}+4 \hat{\mathbf{k}}\), \(3 \hat{\mathbf{i}}+4 \hat{\mathbf{j}}+2 \hat{\mathbf{k}}\) and \(4 \hat{\mathbf{i}}+2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}\) are the vertices of (a) right angled triangle (b) isosceles triangle (c) equilateral triangle (d) collinear

On the \(x y\) plane where \(O\) is the origin, given points, \(A(1,0), B(0,1)\) and \(C(1,1)\). Let \(P, Q\) and \(R\) be moving point on the line \(O A, O B, O C\) respectively such that \(O P=45 t(O A), O Q=60 t(O B), O R=(1-t)(O C)\) with \(t>0\). If the three points \(P_{1} Q\) and \(R\) are collinear, then the value of \(t\) is equal to (a) \(\frac{1}{106}\) (b) \(\frac{7}{187}\) (c) \(\frac{1}{100}\) (d) None of these

In a \(\triangle A B C\) internal angle bisectors \(A I, B I\) and \(C I\) are produced to meet opposite sides in \(A^{\prime}, B^{\prime}\) and \(C^{\prime}\), respectively. Prove that the maximum value of \(\frac{A I \cdot B I \cdot C I}{A A^{\prime} \cdot B B^{\prime} \cdot C C^{\prime}}\) is \(\frac{8}{27}\)

If position vector of points \(A, B\) and \(C\) are respectively \(\hat{i}, \hat{j}\) and \(\hat{\mathbf{k}}\) and \(A B=C X\), then position vector of point \(X\) is \((a)-\hat{i}+\hat{j}+\hat{k}\) (b) \(\hat{i}-\hat{j}+\hat{k}\) (c) \(\hat{\mathbf{i}}+\hat{\mathbf{j}}-\hat{\mathbf{k}}\) (d) \(\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}\)

Let \(a, b\) and \(c\) be three units vectors such thut \(3 a+4 b+5 c=0\) Then which of the following statements is true? (a) a is parallel to \(\mathbf{b}\) (b) a is perpendicular to \(\mathrm{b}\) (c) a is neither parallel nor perpendicular to \(b\) (d) None of the above
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