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Chapter 1: Problem 42

The vectors \(\vec{A}=3 \hat{i}-2 \hat{j}+\hat{k}, \vec{B}=\hat{i}-3 \hat{j}+5 \hat{k}\) and \(\vec{C}=\) \(2 \hat{i}+\hat{j}-4 \hat{k}\) form a triangle, then the triangle is (A) right angled triangle (B) lateral triangle (C) isolated triangle (D) None of these

Short Answer

Expert verified
The triangle formed by the given vectors is a right-angled triangle, option (A).

Step by step solution

01

Compute side vectors (AB, BC, and CA)

Subtract the coordinate vectors: \(\vec{AB} = \vec{B} - \vec{A}\), \(\vec{BC} = \vec{C} - \vec{B}\), and \(\vec{CA} = \vec{A} - \vec{C}\) \(\vec{AB} = (\hat{i}-3 \hat{j}+5 \hat{k}) - (3 \hat{i}-2 \hat{j}+\hat{k}) = -2\hat{i}+\hat{j}+4\hat{k}\) \(\vec{BC} = (2\hat{i}+\hat{j}-4\hat{k}) - (\hat{i}-3\hat{j}+5\hat{k}) = \hat{i}+4\hat{j}-9\hat{k\) \(\vec{CA} = (3 \hat{i}-2 \hat{j}+\hat{k}) - (2 \hat{i}+\hat{j}-4 \hat{k}) = \hat{i}-3\hat{j}+5\hat{k\)
02

Find magnitudes of side vectors

Calculate the magnitudes of the side vectors: \(AB = \sqrt{(-2)^2 + 1^2 + 4^2} = \sqrt{21}\) \(BC = \sqrt{1^2 + 4^2 + (-9)^2} = \sqrt{98}\) \(CA = \sqrt{1^2 + (-3)^2 + 5^2} = \sqrt{35}\)
03

Verify the relationship between side lengths

Before checking the relationship, let's find the squares of the magnitudes: \(AB^2 = 21\) \(BC^2 = 98\) \(CA^2 = 35\) Now, check if the square of the largest side is equal to the sum of the squares of the other two sides. In this case, 98 is the largest, so: \[98 = 21 + 35\] This statement is true, so the triangle is a right-angled triangle. Therefore, the correct answer is (A) right-angled triangle.

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Key Concepts

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

Vectors in Physics
In physics, a vector is a fundamental concept used to describe quantities that possess both magnitude and direction. For example, force, velocity, and acceleration are all vector quantities. Vectors are typically represented by arrows, where the length of the arrow indicates the magnitude of the vector, and the direction of the arrow shows the direction in which the vector is acting.

Mathematically, vectors in a three-dimensional space can be expressed using unit vectors \(\hat{i}\text{, } \hat{j}\text{, and } \hat{k}\). These unit vectors represent the basis vectors in the x, y, and z directions, respectively. A vector such as \(\vec{A}\) can be written as a combination of these unit vectors, each multiplied by a coefficient that represents the component of the vector along that axis.
Magnitude of Vectors
The magnitude of a vector represents its 'size' or 'length' and is a scalar quantity, meaning it has a value without a direction. To find the magnitude of a vector in three-dimensional space, you apply the Pythagorean theorem to its components. The magnitude is given by the square root of the sum of the squares of its components. This is often expressed with the equation \(\|\vec{A}\| = \sqrt{A_x^2 + A_y^2 + A_z^2}\), where \(A_x\), \(A_y\), and \(A_z\) are the components of the vector along the x, y, and z axes, respectively.

Understanding vector magnitudes is crucial since many physical laws, including those for work and kinetic energy, involve vector magnitudes. When solving physics problems, accurately determining vector magnitudes allows you to predict results like displacement, work done, and more.
Right-Angled Triangle
A right-angled triangle is a special kind of triangle with one angle measuring 90 degrees. This property creates a unique relationship between the triangle's sides, known as the Pythagorean theorem. The theorem states that the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. Expressed as an equation, it reads \(c^2 = a^2 + b^2\), where \(c\) represents the length of the hypotenuse, and \(a\) and \(b\) represent the lengths of the other two sides.

In vector problems involving right-angled triangles, we can use this theorem to verify the nature of the triangle by calculating the magnitude of the vectors that form its sides. If the relationship holds true vecfor the squares of the magnitudes, as in the exercise above, the triangle must indeed be right-angled.

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Most popular questions from this chapter

A man starts from \(O\) moves \(500 \mathrm{~m}\) turns by \(60^{\circ}\) and moves \(500 \mathrm{~m}\) again turns by \(60^{\circ}\) and moves \(500 \mathrm{~m}\) and so on. Find the displacement after (i) 5 th turn, (ii) 3 rd turn (A) \(500 \mathrm{~m}, 1000 \mathrm{~m}\) (B) \(500 \mathrm{~m}, 500 \sqrt{3} \mathrm{~m}\) (C) \(1000 \mathrm{~m}, 500 \sqrt{3} \mathrm{~m}\) (D) None of these

A screw gauge with a pitch of \(0.5 \mathrm{~mm}\) and a circular scale with 50 divisions is used to measure the thickness of a thin sheet of Aluminium. Before starting the measurement, it is found that when the two jaws of the screw gauge are brought in contact, the \(45^{\text {th }}\) division coincides with the main scale line and that the zero of the main scale is barely visible. What is the thickness of the sheet if the main scale reading is \(0.5 \mathrm{~mm}\) and the \(25^{\text {th }}\) division coincides with the main scale line? [2016] (A) \(0.80 \mathrm{~mm}\) (B) \(0.70 \mathrm{~mm}\) (C) \(0.50 \mathrm{~mm}\) (D) \(0.75 \mathrm{~mm}\)

If \(\vec{a}\) and \(\vec{b}\) are two vectors then the value of \((\vec{a}+\vec{b}) \times(\vec{a}-\vec{b})\) is (A) \(2(\vec{b} \times \vec{a})\) (B) \(-2(\vec{b} \times \vec{a})\) (C) \(\vec{b} \times \vec{a}\) (D) \(\vec{a} \times \vec{b}\)

If we change unit of a physical quantity then (A) its dimension changes. (B) its dimension remain same. (C) it may change or may not change. (D) its magnitude changes.

The pitch of a screw gauge is \(1 \mathrm{~mm}\) and there are 100 divisions on its circular scale. When nothing is put in between its jaws, the zero of the circular scale lies 4 divisions below the reference line. When a steel wire is placed between the jaws, two main scale divisions are clearly visible and 67 divisions on the circular scale are observed. The diameter of the wire is (A) \(2.71 \mathrm{~mm}\) (B) \(2.67 \mathrm{~mm}\) (C) \(2.63 \mathrm{~mm}\) (D) \(2.65 \mathrm{~mm}\)
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