# Inter 1st Year Maths 1A Products of Vectors Solutions Ex 5(c)

Practicing the Intermediate 1st Year Maths 1A Textbook Solutions Inter 1st Year Maths 1A Products of Vectors Solutions Exercise 5(c) will help students to clear their doubts quickly.

## Intermediate 1st Year Maths 1A Products of Vectors Solutions Exercise 5(c)

I.

Question 1.
Compute $$[\overline{\mathbf{i}}-\overline{\mathbf{j}} \overline{\mathbf{j}}-\overline{\mathbf{k}} \overline{\mathbf{k}}-\overline{\mathbf{i}}]$$
Solution:

Question 2.
If $$\overline{\mathbf{a}}=\overline{\mathbf{i}}-2 \overline{\mathbf{j}}-3 \overline{\mathbf{k}}, \overline{\mathbf{b}}=2 \overline{\mathbf{i}}+\overline{\mathbf{j}}-\overline{\mathbf{k}}$$, $$\bar{c}=\bar{i}+3 \bar{j}-2 \bar{k}$$, then compute $$\overline{\mathbf{a}} \cdot(\overline{\mathbf{b}} \times \overline{\mathbf{c}})$$.
Solution:

Question 3.
If $$\bar{a}$$ = (1, -1, -6), $$\bar{b}$$ = (1, -3, 4) and $$\bar{c}$$ = (2, -5, 3), then compute the following
(i) $$\overline{\mathbf{a}} \cdot(\bar{b} \times \bar{c})$$
(ii) $$\overline{\mathbf{a}} \times(\overline{\mathbf{b}} \times \overline{\mathbf{c}})$$
(iii) $$(\overline{\mathbf{a}} \times \overline{\mathbf{b}}) \times \overline{\mathbf{c}}$$
Solution:

Question 4.
Simplify the following.
(i) $$(\bar{i}-2 \bar{j}+3 \bar{k}) \times(2 i+j-\bar{k}) \cdot(\bar{j}+\bar{k})$$
(ii) $$(2 \bar{i}-3 \bar{j}+\bar{k}) \cdot(\bar{i}-\bar{j}+2 \bar{k}) \cdot(2 \bar{i}+\bar{j}+\bar{k})$$
Solution:

Question 5.
Find the volume of the parallelopiped having coterminous edges.
$$\overline{\mathbf{i}}+\overline{\mathbf{j}}+\overline{\mathbf{k}}, \overline{\mathbf{i}}-\overline{\mathbf{j}}$$ and $$\overline{\mathbf{i}}+\mathbf{2} \overline{\mathbf{j}}-\overline{\mathbf{k}}$$
Solution:
Let $$\overline{\mathrm{a}}=\overline{\mathrm{i}}+\overline{\mathrm{j}}+\overline{\mathrm{k}}, \overline{\mathrm{b}}=\overline{\mathrm{i}}-\overline{\mathrm{j}}$$ and $$\overline{\mathrm{c}}=\overline{\mathrm{i}}+2 \overline{\mathrm{j}}-\overline{\mathrm{k}}$$
Volume of the parallelopiped = $$[(\bar{a} \bar{b} \bar{c})]$$
= $$\left|\begin{array}{ccc} 1 & 1 & 1 \\ 1 & -1 & 0 \\ 1 & 2 & -1 \end{array}\right|$$
= 1(1 – 0) – 1(-1 – 0) + 1(2 + 1)
= 1 + 1 + 3
= 5 cubic units.

Question 6.
Find t for which the vectors $$\mathbf{2} \overline{\mathbf{i}}-\mathbf{3} \overline{\mathbf{j}}+\overline{\mathbf{k}}$$, $$\bar{i}+2 \mathbf{j}-3 \bar{k}$$ and $$\bar{j}-t \bar{k}$$ are coplanar.
Solution:

Question 7.
For non-coplanar vectors, $$\bar{a}$$, $$\bar{b}$$ and $$\bar{c}$$ determine p for which the vector $$\overline{\mathbf{a}}+\overline{\mathbf{b}}+\overline{\mathbf{c}}, \overline{\mathbf{a}}+\mathbf{p} \overline{\mathbf{b}}+\mathbf{2} \overline{\mathbf{c}}$$ and $$-\overline{\mathbf{a}}+\overline{\mathbf{b}}+\overline{\mathbf{c}}$$ are coplanar.
Solution:

Question 8.
Determine λ, for which the volume of the parallelopiped having coterminous edges $$\bar{i}+\bar{j}$$, $$3 \overline{\mathbf{i}}-\overline{\mathbf{j}}$$ and $$3 \bar{j}+\lambda \bar{k}$$ is 16 cubic units.
Solution:

Question 9.
Find the volume of the tetrahedron having the edges $$\overline{\mathbf{i}}+\overline{\mathbf{j}}+\overline{\mathbf{k}}, \quad \mathbf{i}-\overline{\mathbf{j}}$$ and $$\bar{i}+2 \bar{j}+\bar{k}$$.
Solution:

Question 10.
Let $$\bar{a}$$, $$\bar{b}$$ and $$\bar{c}$$ be non-coplanar vectors and $$\bar{\alpha}=\bar{a}+2 \bar{b}+3 c, \quad \bar{\beta}=2 \bar{a}+\bar{b}-2 c$$ and $$\bar{\gamma}=3 \bar{a}-7 \bar{c}$$, then find $$\left[\begin{array}{lll} \bar{\alpha} & \bar{\beta} & \bar{\gamma} \end{array}\right]$$.
Solution:

Question 11.
Let $$\bar{a}$$, $$\bar{b}$$ and $$\bar{c}$$ be non-coplanar vectors. If $$|2 \bar{a}-\bar{b}+3 \bar{c}|, \bar{a}+\bar{b}-2 \bar{c},|\bar{a}+\bar{b}-3 \bar{c}|$$ = $$\lambda[\overline{\mathbf{a}} \overline{\mathbf{b}} \overline{\mathbf{c}}]$$, then find the value of λ.
Solution:

Question 12.
Let $$\bar{a}$$, $$\bar{b}$$ and $$\bar{c}$$ be non-coplanar vectors, if $$\left[\begin{array}{lll} \bar{a}+2 \bar{b} & 2 \bar{b}+\bar{c} & 5 \bar{c}+\bar{a} \end{array}\right]$$ = $$\lambda\left[\begin{array}{lll} \bar{a} & \bar{b} & \bar{c} \end{array}\right]$$, then find λ.
Solution:

Question 13.
If $$\bar{a}$$, $$\bar{b}$$ and $$\bar{c}$$ are non-coplanar vectors, then find the value of $$\frac{(\bar{a}+2 \bar{b}-\bar{c}) \cdot[(\bar{a}-\bar{b}) \times(\bar{a}-\bar{b}-\bar{c})]}{[\bar{a} \bar{b} \bar{c}]}$$
Solution:

Question 14.
If $$\bar{a}$$, $$\bar{b}$$ and $$\bar{c}$$ are mutually perpendicular unit vectors, then find the value of $$\left[\begin{array}{lll} \bar{a} & \bar{b} & \bar{c} \end{array}\right]^{2}$$.
Solution:

Question 15.
$$\overline{\mathbf{a}}, \overline{\mathbf{b}}, \overline{\mathbf{c}}$$ are non-zero vectors and $$\overline{\mathbf{a}}$$ is perpendicular to both $$\overline{\mathbf{b}}$$ and $$\overline{\mathbf{c}}$$. If $$|\overline{\mathbf{a}}|$$ = 2, $$|\overline{\mathbf{b}}|$$ = 3, $$|\overline{\mathbf{c}}|$$ = 4 and $$(\bar{b}, \bar{c})=\frac{2 \pi}{3}$$, then find $$|[\bar{a} \bar{b} \bar{c}]|$$.
Solution:

Question 16.
If $$\overline{\mathbf{a}}, \overline{\mathbf{b}}, \overline{\mathbf{c}}$$ are unit coplanar vectors, then find $$\left[\begin{array}{lll} 2 \bar{a}-\bar{b} & 2 \bar{b}-\bar{c} & 2 \bar{c}-\bar{a} \end{array}\right]$$
Solution:

II.

Question 1.
If $$\left[\begin{array}{lll} \bar{b} & \bar{c} & \bar{d} \end{array}\right]+\left[\begin{array}{lll} \bar{c} & \bar{a} & \bar{d} \end{array}\right]+\left[\begin{array}{lll} \bar{a} & \bar{b} & \bar{d} \end{array}\right]$$ = $$\left[\begin{array}{lll} \overline{\mathbf{a}} & \overline{\mathbf{b}} & \overline{\mathbf{c}} \end{array}\right]$$ then show that the points with position vectors $$\overline{\mathbf{a}}, \overline{\mathbf{b}}, \overline{\mathbf{c}}$$ and $$\bar{d}$$ are coplanar.
Solution:

Question 2.
If $$\overline{\mathbf{a}}, \overline{\mathbf{b}}$$ and $$\overline{\mathbf{c}}$$ non-coplanar vectors, then prove that the four points with position vectors $$2 \bar{a}+3 \bar{b}-\bar{c}$$, $$\overline{\mathrm{a}}-2 \overline{\mathrm{b}}+3 \overline{\mathrm{c}}, 3 \overline{\mathrm{a}}+4 \overline{\mathrm{b}}-2 \overline{\mathrm{c}}$$ and $$\bar{a}-6 \bar{b}+6 \bar{c}$$ are coplanar.
Solution:
Suppose A, B, C, D are the given points.

The vectors $$\overrightarrow{\mathrm{AB}}, \overrightarrow{\mathrm{AC}}, \overrightarrow{\mathrm{AD}}$$ are coplanar.
The given points A, B, C, D are coplanar.

Question 3.
$$\overline{\mathbf{a}}, \overline{\mathbf{b}}$$ and $$\overline{\mathbf{c}}$$ are non-zero and non- collinear vectors and θ ≠ 0, is the angle between $$\overline{\mathbf{b}}$$ and $$\overline{\mathbf{c}}$$. If $$(\overline{\mathbf{a}} \times \overline{\mathbf{b}}) \times \overline{\mathbf{c}}$$ = $$\frac{1}{3}|\bar{b}||\bar{c}|\bar{a}|$$, then find sin θ.
Solution:

Question 4.
Find the volume of the tetrahedron whose vertices are (1, 2, 1), (3, 2, 5), (2, -1, 0) and (-1, 0, 1).
Solution:
Let ‘O’ be the given A, B, C, D be the vertices of the ten tetrahedrons. Then

Question 5.
Show that $$(\bar{a}+\bar{b}) \cdot(\bar{b}+\bar{c}) \times(\bar{c}+\bar{a})$$ = $$2\left[\begin{array}{lll} \bar{a} & \bar{b} & \bar{c} \end{array}\right]$$
Solution:

Question 6.
Show that equation of the plane passing through the points with position vectors. $$3 \bar{i}-5 \bar{j}-\overline{\mathbf{k}},-\overline{\mathbf{i}}+5 \bar{j}+7 \overline{\mathbf{k}}$$ and parallel to the vector $$3 \overline{\mathbf{i}}-\overline{\mathbf{j}}+7 \overline{\mathbf{k}}$$ is 3x + 2y – z = 0.
Solution:
The given plane passes through the points A, B (i.e.,) $$3 \bar{i}-5 \bar{j}-\overline{\mathbf{k}},-\overline{\mathbf{i}}+5 \bar{j}+7 \overline{\mathbf{k}}$$ and parallel to the vector $$3 \overline{\mathbf{i}}-\overline{\mathbf{j}}+7 \overline{\mathbf{k}}$$

= x(70 + 8) – y(-28 – 24) + z(4 – 30)
= 78x + 52y – 26z
= 26(3x + 2y – z)
$$\left[\begin{array}{lll} \bar{a} & \bar{b} & \bar{c} \end{array}\right]$$ = $$\left|\begin{array}{rrr} 3 & -5 & -1 \\ -4 & 10 & 8 \\ 3 & -1 & 7 \end{array}\right|$$
= 3(70 + 8) + 5(-28 – 24) – 1(4 – 30)
= 234 – 260 + 26
= 0
Equation of the required plane is 26(3x + 2y – z) = 0
i.e., 3x + 2y – z = 0

Question 7.
Prove that $$\overline{\mathbf{a}} \times[\overline{\mathbf{a}} \times(\overline{\mathbf{a}} \times \overline{\mathbf{b}})]$$ = $$(\overline{\mathbf{a}} \cdot \overline{\mathbf{a}})(\overline{\mathbf{b}} \times \overline{\mathbf{a}})$$
Solution:

Question 8.
If $$\overline{\mathbf{a}}, \overline{\mathbf{b}}, \overline{\mathbf{c}}$$ and $$\bar{d}$$ are coplanar vectors, then show that $$(\bar{a} \times \bar{b}) \times(\bar{c} \times \bar{d})=0$$.
Solution:
$$\overline{\mathbf{a}}, \overline{\mathbf{b}}, \overline{\mathbf{c}}$$ and $$\bar{d}$$ are coplanar
⇒ $$\overline{\mathbf{a}} \times \overline{\mathbf{b}}$$ is perpendicular to the plane π.
similarly $$\bar{c} \times \bar{d}$$ is perpendicular to the plane π.
$$\bar{a} \times \bar{b}$$ and $$\bar{c} \times \bar{d}$$ are parallel vectors.
⇒ $$(\bar{a} \times \bar{b}) \times(\bar{c} \times \bar{d})$$ = 0.

Question 9.
Show that $$(\overline{\mathrm{a}} \times \overline{\mathrm{b}}) \times(\overline{\mathrm{a}} \times \overline{\mathrm{c}}) \cdot \overline{\mathrm{d}}=[\overline{\mathrm{a}} \overline{\mathrm{b}} \overline{\mathrm{c}}](\overline{\mathrm{a}} \cdot \overline{\mathrm{d}})$$.
Solution:

Question 10.
Show that $$\bar{a} \cdot[(\bar{b}+\bar{c}) \times(\bar{a}+\bar{b}+\bar{c})]=0$$
Solution:

Question 11.
Find λ in order that the four points A(3, 2, 1), B(4, λ, 5), C(4, 2, -2) and D(6, 5, -1) be coplanar.
Solution:

Question 12.
Find the vector equation of the plane passing through the intersection of planes $$\bar{r} \cdot(2 \bar{i}+2 \bar{j}-3 \bar{k})=7, \bar{r} \cdot(2 \bar{i}+5 \bar{j}+3 \bar{k})=9$$ and through the point (2, 1, 3)
Solution:

Question 13.
Find the equation of the plane passing through (a, b, c) and parallel to the plane $$\bar{r} \cdot(\bar{i}+\bar{i}+\bar{k})=2$$.
Solution:
Given equation plane is $$\bar{r} \cdot(\bar{i}+\bar{i}+\bar{k})=2$$
Let $$\bar{r}=x \bar{i}+y \bar{j}+z \bar{k}$$
∴ $$\bar{r} \cdot(\bar{i}+\bar{i}+\bar{k})=2$$
$$(x \bar{i}+y \bar{j}+z \bar{k}) \cdot(i+j+k)=2$$
x + y + z = 2
Required plane equation is x + y + z = k …….(1)
Equation (1) passes through (a, b, c)
∴ a + b + c = k
Substitute ‘k’ in equation (1)
∴ x + y + z = a + b + c

Question 14.
Find the shortest distance between the lines $$\bar{r}=6 \bar{i}+2 \bar{j}+2 \bar{k}+\lambda, \bar{i}-2 \bar{j}+2 \bar{k}$$ and $$\bar{r}=-4 \bar{j}-\bar{k}+\mu=3 \bar{j}-2 \bar{j}-2 \bar{k}$$.
Solution:
The first line passes through point A(6, 2, 2) and is parallel to the vector b = i – 2j + 2k.
Second line passes through the point C(-4, 0, -1) and is parallel to the vector d = 3i – 2j – 2k

Question 15.
Find the equation of the plane passing through the line of intersection of the planes $$\bar{r} \cdot(\bar{i}+\bar{j}+\bar{k})=1$$ and $$\bar{r} \cdot(2 \bar{i}+3 \bar{i}-\bar{k})+4=0$$ and parallel to X-axis.
Solution:
Given the equation of planes are

Since it is parallel to X-axis.

Question 16.
Prove that the four points $$4 \bar{i}+5 \bar{j}+\bar{k}$$, $$-(\overline{\mathbf{j}}+\overline{\mathbf{k}}), 3 \overline{\mathbf{i}}+9 \overline{\mathbf{j}}+4 \overline{\mathbf{k}}$$ and $$-4 \bar{i}+4 \bar{j}+4 \bar{k}$$ are coplanar.
Solution:
Let ‘O’ be the origin. A, B, C, D be the given points. Then

Question 17.
If $$\bar{a}, \bar{b}, \bar{c}$$ are non – copianar, then show that the vectors $$\overline{\mathbf{a}}-\overline{\mathbf{b}}, \overline{\mathbf{b}}+\overline{\mathbf{c}}$$, $$\overline{\mathbf{c}}+\overline{\mathbf{a}}$$ are coplanar.
Solution:

Question 18.
If $$\overline{\mathbf{a}}, \overline{\mathbf{b}}, \overline{\mathbf{c}}$$ are the position vectors of the points A, B and C respectively, then prove that the vector $$\overline{\mathbf{a}} \times \overline{\mathbf{b}}+\overline{\mathbf{b}} \times \overline{\mathbf{c}}+\overline{\mathbf{c}} \times \overline{\mathbf{a}}$$ is perpendicular to the plane of ∆ABC.
Solution:

III.

Question 1.
Show that $$(\bar{a} \times(\bar{b} \times \bar{c}) \times \bar{c})=(\bar{a} \cdot \bar{c})(\bar{b} \times \bar{c})$$ and $$(\overline{\mathbf{a}} \times \overline{\mathbf{b}}) \cdot(\overline{\mathbf{a}} \times \overline{\mathbf{c}})+(\overline{\mathbf{a}} \cdot \overline{\mathbf{b}})(\overline{\mathbf{a}} \cdot \overline{\mathbf{c}})$$ = $$(\overline{\mathbf{a}} \cdot \overline{\mathbf{a}})(\overline{\mathbf{b}} \cdot \overline{\mathbf{c}})$$
Solution:

Question 2.
If A = (1, -2, -1), B = (4, 0, -3), C = (1, 2, -1) and D = (2, -4, -5), find the distance between AB and CD.
Solution:

Question 3.
If $$\overline{\mathbf{a}}=\overline{\mathbf{i}}-2 \overline{\mathbf{j}}+\overline{\mathbf{k}}, \overline{\mathbf{b}}=2 \overline{\mathbf{i}}+\overline{\mathbf{j}}+\overline{\mathbf{k}}$$ and $$\overline{\mathbf{c}}=\overline{\mathbf{i}}+2 \overline{\mathbf{j}}-\overline{\mathbf{k}}$$, find $$\bar{a} \times(\bar{b} \times \bar{c})$$ and $$|(\overline{\mathbf{a}} \times \overline{\mathbf{b}}) \times \overline{\mathbf{c}}|$$.
Solution:

Question 4.
If $$\overline{\mathbf{a}}=\overline{\mathbf{i}}-2 \overline{\mathbf{j}}-3 \overline{\mathbf{k}}, \overline{\mathbf{b}}=2 \overline{\mathbf{i}}+\overline{\mathbf{j}}-\overline{\mathbf{k}}$$ and $$\bar{c}=\bar{i}+3 \bar{j}-2 \bar{k}$$, verift that $$\overline{\mathbf{a}} \times(\overline{\mathbf{b}} \times \overline{\mathbf{c}}) \neq(\overline{\mathbf{a}} \times \overline{\mathbf{b}}) \times \overline{\mathbf{c}}$$
Solution:

From (1) and (2), we get
$$\overline{\mathbf{a}} \times(\overline{\mathbf{b}} \times \overline{\mathbf{c}}) \neq(\overline{\mathbf{a}} \times \overline{\mathbf{b}}) \times \overline{\mathbf{c}}$$
i.e., vector multiplication is not associative.

Question 5.
If $$\overline{\mathbf{a}}=2 \overline{\mathbf{i}}+\overline{\mathbf{j}}-3 \overline{\mathbf{k}}, \overline{\mathbf{b}}=\overline{\mathbf{i}}-\mathbf{2} \mathbf{j}+\overline{\mathbf{k}}$$, $$\bar{c}=-\bar{i}+\bar{j}-4 \bar{k}$$ and $$\overline{\mathbf{d}}=\overline{\mathbf{i}}+\overline{\mathbf{j}}+\overline{\mathbf{k}}$$, then compute $$|(\overline{\mathrm{a}} \times \overline{\mathrm{b}}) \times(\overline{\mathrm{c}} \times \overline{\mathrm{d}})|$$
Solution:

Question 6.
If A = (1, a, a2), B = (1, b, b2) and C = (1, c, c2) are non-coplanar vectors and $$\left|\begin{array}{lll} a & a^{2} & 1+a^{3} \\ b & b^{2} & 1+b^{3} \\ c & c^{2} & 1+c^{3} \end{array}\right|$$ = 0, then show that abc + 1 = 0
Solution:
$$\bar{A}, \bar{B}, \bar{C}$$ are non-coplanar vectors.
∆ = $$\left|\begin{array}{lll} 1 & a & a^{2} \\ 1 & b & b^{2} \\ 1 & c & c^{2} \end{array}\right|$$ ≠ 0 ………..(1)

∆ + (abc) ∆ = 0; ∆(1 + abc) = 0
∆ ≠ 0 ⇒ 1 + abc = 0

Question 7.
If $$\overline{\mathbf{a}}, \overline{\mathbf{b}}, \overline{\mathbf{c}}$$ are non-zero vectors, then $$|(\overline{\mathbf{a}} \times \mathbf{b} \cdot \overline{\mathbf{c}})|=|\overline{\mathbf{a}}||\mathbf{b}||\overline{\mathbf{c}}|$$ $$\Leftrightarrow \overline{\mathbf{a}} \cdot \overline{\mathbf{b}}=\overline{\mathbf{b}} \cdot \overline{\mathbf{c}}=\overline{\mathbf{c}} \cdot \overline{\mathbf{a}}=\mathbf{0}$$
Solution:

Question 8.
If $$\overline{\mathbf{a}}=\overline{\mathbf{i}}-\mathbf{2} \overline{\mathbf{j}}+3 \overline{\mathbf{k}}, \quad \mathbf{b}=2 \overline{\mathbf{i}}+\overline{\mathbf{j}}+\overline{\mathbf{k}}$$, $$\bar{c}=\overline{\mathbf{i}}+\overline{\mathbf{j}}+2 \overline{\mathbf{k}}$$ then find $$|(\bar{a} \times \bar{b}) \times \bar{c}|$$ and $$|\overline{\mathbf{a}} \times(\mathbf{b} \times \mathbf{c})|$$.
Solution:

Question 9.
If $$|\bar{a}|=1,|\bar{b}|=1,|\bar{c}|=2$$ and $$\bar{a} \times(a \times \bar{c})+\bar{b}=0$$ then find the angle between $$\bar{a}$$ and $$\bar{c}$$.
Solution:

Question 10.
Let $$\overline{\mathbf{a}}=\overline{\mathbf{i}}-\overline{\mathbf{k}}, \quad \overline{\mathbf{b}}=\mathbf{x} \overline{\mathbf{i}}+\overline{\mathbf{j}}+(\mathbf{1}-\mathbf{x}) \overline{\mathbf{k}}$$ and $$\bar{c}=y \bar{i}+x \bar{j}+(1+x-y) \bar{k}$$, prove that the scalar triple product $$\left[\begin{array}{lll} \bar{a} & \bar{b} & \bar{c} \end{array}\right]$$ is independent of both x and y.
Solution:

Question 11.
Let $$\overline{\mathbf{b}}=\mathbf{2} \overline{\mathbf{i}}+\overline{\mathbf{j}}-\overline{\mathbf{k}}, \overline{\mathbf{c}}=\overline{\mathbf{i}}+\mathbf{3} \overline{\mathbf{k}}$$. If $$\overline{\mathrm{a}}$$ is a unit vector then find the maximum value of $$\left[\begin{array}{lll} \overline{\mathbf{a}} & \overline{\mathbf{b}} & \bar{c} \end{array}\right]$$.
Solution:
Let $$\bar{a}=x \bar{i}+y \bar{j}+z \bar{k}$$ and x2 + y2 + z2 = 1
∵ $$\overline{\mathrm{a}}$$ unit vector

Question 12.
Let $$\overline{\mathrm{a}}=\overline{\mathrm{i}}-\overline{\mathrm{i}}, \overline{\mathrm{b}}=\overline{\mathrm{i}}-\overline{\mathrm{k}}, \overline{\mathrm{c}}=\overline{\mathrm{k}}-\overline{\mathrm{i}}$$ Find unit vector $$\bar{d}$$ such that $$\overline{\mathrm{a}} \cdot \overline{\mathrm{d}}=0=[\bar{b} \overline{\mathrm{c}} \overline{\mathrm{d}}]$$
Solution: