The vector a=α i+2j+β k lies in the plane of the vectors b=i+j and c=j+k and bisects the angle between b and c. Then which of the following gives possible values of α and β?

D. α =1, β =1. Step 1: |b|=|c|=√(2), so the bisector is along b+c=i+2j+k. Step 2: a must be a multiple of i+2j+k; its j component 2 makes the multiple 1. Step 3: So α =1 and β =1. Check: a=i+2j+k gives a⋅b=3=a⋅c, equal angles with two vectors of equal length. Trap: (A) 2i+2j+2k makes equal angles too, but it is not in the plane of b and c: x(1,1,0)+y(0,1,1)=(2,2,2) needs x=y=2 and x+y=2.

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The vector $\overset{a}{^} = α \hat{i} + 2 \hat{j} + β \hat{k}$ lies in the plane of the vectors $\hat{b} = \hat{i} + \hat{j}$ and $\overset{c}{^} = \hat{j} + \hat{k}$ and bisects the angle between $\hat{b}$ and $\overset{c}{^}$ . Then which of the following gives possible values of $α$ and $β$ ?

Answer (Detailed Solution Below)View answer ▾

Answer: Option D — $α = 1$ , $β = 1$

Detailed Solution

$\hat{b} = ∣ \overset{c}{^} ∣ = 2$ , so the bisector is along $\hat{b} + \overset{c}{^} = \hat{i} + 2 \hat{j} + \hat{k}$ .

$\overset{a}{^}$ must be a multiple of $\hat{i} + 2 \hat{j} + \hat{k}$ ; its $\hat{j}$ component 2 makes the multiple 1.

So $α = 1$ and $β = 1$ .

✓ Check

\overset{a}{^} = \hat{i} + 2 \hat{j} + \hat{k}

gives

\overset{a}{^} \cdot \hat{b} = 3 = \overset{a}{^} \cdot \overset{c}{^}

, equal angles with two vectors of equal length.

⚠ Common trap

(A)

2 \hat{i} + 2 \hat{j} + 2 \hat{k}

makes equal angles too, but it is not in the plane of

\hat{b}

and

\overset{c}{^}

x (1, 1, 0) + y (0, 1, 1) = (2, 2, 2)

needs

x = y = 2

and

x + y = 2

Quick tip

Bisector of equal-length vectors: $b + c = \hat{i} + 2 \hat{j} + \hat{k}$

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Vectors — Preparation

This question tests your understanding of Vectors, a key topic in the Quantitative Aptitude section. The detailed solution above uses a step-by-step approach to show the reasoning, and where available, a shortcut that helps you solve similar questions in under 90 seconds during the actual exam.

A vector is a quantity with both magnitude and direction, such as displacement or force. This topic covers types of vectors (zero, unit, collinear, coplanar and position vectors), vector addition, multiplication by a scalar, the section formula, and the components of a vector along the i, j and k axes. It also covers the scalar (dot) product and its use in finding the angle between two vectors and the projection of one vector on another, the vector (cross) product and its use in finding areas of parallelograms and triangles, and the scalar triple product for the volume of a parallelepiped and the condition for coplanarity. Remember that two vectors are perpendicular when their dot product is zero, parallel when their cross product is the zero vector, and three vectors are coplanar when their scalar triple product is zero. Practise the vectors questions below, each with a step-by-step solution.

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The vector a^=αi^+2j^​+βk^ lies in the plane of the vectors b^=i^+j^​ and c^=j^​+k^ and bisects the angle between b^ and c^. Then which of the following gives possible values of α and β?

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