Let the position vectors of the points P, Q, R and S be
\(\vec{a}=\hat{i}+2\hat{j}-5\hat{k}\), \(\vec{b}=3\hat{i}+6\hat{j}+3\hat{k}\), \(\vec{c}=\frac{17}{5}\hat{i}+\frac{16}{5}\hat{j}+7\hat{k}\) and \(\vec{d}=2\hat{i}+\hat{j}+\hat{k}\)
respectively. Then which of the following statements is true?
Let the position vectors of the points P, Q, R and S be
\(\vec{a}=\hat{i}+2\hat{j}-5\hat{k}\), \(\vec{b}=3\hat{i}+6\hat{j}+3\hat{k}\), \(\vec{c}=\frac{17}{5}\hat{i}+\frac{16}{5}\hat{j}+7\hat{k}\) and \(\vec{d}=2\hat{i}+\hat{j}+\hat{k}\)
respectively. Then which of the following statements is true?
- The points P,Q,R and S are NOT coplanar
- \(\frac{\vec{b}+2\vec{d}}{3}\) is the position vector of a point that divides PR internally in the ratio 5:4
- \(\frac{\vec{b}+2\vec{d}}{3}\) is the position vector of a point that divides PR externally in the ratio 5:4
- The square of the magnitude of the vector \(\vec{b}\times \vec{d}\) is 95
The Correct Option is B
Approach Solution - 1
Given :
The position vector of point P is given by: \(\vec{P} = \hat{i} + 2\hat{j} - 5\hat{k}\)
The position vector of point R is given by: \(\vec{R} = \frac{17}{5}\hat{i} + \frac{16}{5}\hat{j} + 7\hat{k}\)
Now, let's find the position vector of the point that divides PR internally in the ratio 5:4. This can be done using the section formula:
4M=5+45R+4P
\(\vec{M} = \frac{5\left(\frac{17}{5}\hat{i} + \frac{16}{5}\hat{j} + 7\hat{k} \right) + 4\left(\hat{i} + 2\hat{j} - 5\hat{k} \right)}{9}\)
\(\vec{M} = \frac{17\hat{i} + 16\hat{j} + 35\hat{k} + 4\hat{i} + 8\hat{j} - 20\hat{k}}{9}\)
\(\vec{M} = \frac{21\hat{i} + 24\hat{j} + 15\hat{k}}{9}\)
\(\vec{M} = \frac{7\hat{i} + 8\hat{j} + 5\hat{k}}{3}\)
This is the same as 3b+2d, which confirms that the option (B) is indeed correct. It represents a point that divides PR internally in the ratio 5:4.
So, the correct option is (B): \(\frac{\vec{b}+2\vec{d}}{3}\) is the position vector of a point that divides PR internally in the ratio 5:4
Approach Solution -2
Given :
P(1, 2, -5), Q(3, 6, 3), \(R(\frac{17}{5},\ \frac{16}{5},\ 7)\), S(2, 1, 1)
\(\frac{\vec{b}+2\vec{d}}{3}=\frac{7\hat{i}+8\hat{j}+5\hat{k}}{3}\)
\(⇒\frac{17\lambda}{5}+1=\frac{7}{3}(\lambda+1)\)
\(⇒51λ+15=35λ+35\)
\(⇒16λ=20\)
\(⇒λ=\frac{5}{4}\)
So, the correct option is (B) : \(\frac{\vec{b}+2\vec{d}}{3}\) is the position vector of a point that divides PR internally in the ratio 5:4
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Concepts Used:
Linear Inequalities in One Variable
The graph of a linear inequality in one variable is a number line. We can use an open circle for < and > and a closed circle for ≤ and ≥.
Inequalities that have the same solution are commonly known as equivalents. There are several properties of inequalities as well as the properties of equality. All the properties below are also true for inequalities including ≥ and ≤.
The addition property of inequality says that adding the same number to each side of the inequality gives an equivalent inequality.
If x>y, then x+z>y+z If x>y, then x+z>y+z
If x<y, then x+z<y+z If x<y, then x+z<y+z
The subtraction property of inequality tells us that subtracting the same number from both sides of an inequality produces an equivalent inequality.
If x>y, then x−z>y−z If x>y, then x−z>y−z
If x<y, then x−z<y−z Ifx<y, then x−z<y−z
The multiplication property of inequality tells us that multiplication on both sides of an inequality with a positive number gives an equivalent inequality.
If x>y and z>0, then xz>yz If x>y and z>0, then xz>yz
If x<y and z>0, then xz<yz If x<y and z>0,then xz<yz