Let x-y and z be unit vectors such that x-y-z-a- x-y-z-b- x-y-z-c-a-x-32- a-y-74 and -a-2-Then which of the following options is-are CORRECT -

X+y+z=a Taking dot product with a, ⇒a·x+a·y+a·z=a·a=|a|^2=4 ⇒a·z=4 32 74=34 Now, taking dot product with x and y simultaneously, x· (x+y+z)=x·a=a·x=32 ⇒x·y+x·z ...

You visited us 6 times! Enjoying our articles? Unlock Full Access!

Byju's Answer

Standard XIII

Mathematics

Vector Triple Product

Let x,y and z...

Question

Let $\to x, \to y$ and $\to z$ be unit vectors such that $\to x + \to y + \to z = \to a, \to x \times (\to y \times \to z) = \to b, (\to x \times \to y) \times \to z = \to c$ ,
$\to a \cdot \to x = \frac{3}{2}, \to a \cdot \to y = \frac{7}{4}$ and $| \to a | = 2$ .

Then which of the following option(s) is/are CORRECT ?

\to a \cdot \to z = \frac{3}{2}

No worries! We‘ve got your back. Try BYJU‘S free classes today!

\to y \cdot \to z = 0

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses

\to z = \frac{4}{3} (\to c - \to b)

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses

\to y = 4 \to c

No worries! We‘ve got your back. Try BYJU‘S free classes today!

Open in App

Solution

The correct option is C $\to z = \frac{4}{3} (\to c - \to b)$

$\to x + \to y + \to z = \to a$
Taking dot product with $\to a$ ,
$\Rightarrow \to a \cdot \to x + \to a \cdot \to y + \to a \cdot \to z = \to a \cdot \to a = | \to a |^{2} = 4$
$\Rightarrow \to a \cdot \to z = 4 - \frac{3}{2} - \frac{7}{4} = \frac{3}{4}$

Now, taking dot product with $\to x$ and $\to y$ simultaneously,
$\to x \cdot (\to x + \to y + \to z) = \to x \cdot \to a = \to a \cdot \to x = \frac{3}{2}$
$\Rightarrow \to x \cdot \to y + \to x \cdot \to z = \frac{1}{2} \dots (1)$

Similarly, $\to y \cdot (\to x + \to y + \to z) = \to y \cdot \to a = \to a \cdot \to y = \frac{7}{4}$
$\Rightarrow \to x \cdot \to y + \to y \cdot \to z = \frac{3}{4} \dots (2)$

$| \to x + \to y + \to z |^{2} = | \to a |^{2}$
$\Rightarrow 3 + 2 (\to x \cdot \to y + \to y \cdot \to z + \to z \cdot \to x) = 4$
$\Rightarrow \to x \cdot \to y + \to y \cdot \to z + \to z \cdot \to x = \frac{1}{2} \dots (3)$

From $(1)$ and $(3)$ , we get
$\to y \cdot \to z = 0$
From $(2)$ and $(3)$ , we get
$\to x \cdot \to z = \frac{- 1}{4}$
From $(3)$ , we get
$\to x \cdot \to y = \frac{3}{4}$

Now, $\to x \times (\to y \times \to z) = \to b$
$\Rightarrow (\to x \cdot \to z) \to y - (\to x \cdot \to y) \to z = \to b$
$\Rightarrow \frac{- 1}{4} \to y - \frac{3}{4} \to z = \to b \dots (4)$

Also, $(\to x \times \to y) \times \to z = \to c$
$\Rightarrow \frac{- 1}{4} \to y = \to c$
$\Rightarrow \to y = - 4 \to c$
So, from $(4)$ , $\to z = \frac{4}{3} (\to c - \to b)$

Suggest Corrections

Similar questions

Q. If

\to x, \to y, \to z

are units vectors such that

\to x + \to y + \to z = \to a

(\to x \times \to y) \times \to z = \to b

\to a \cdot \to x = \frac{3}{2}, \to a \cdot \to y = \frac{7}{4}

and

| \to a | = 2,

then which of the following is FALSE?

Q. Let

\to x, \to y

and

\to z

be unit vectors such that

\to x + \to y + \to z = \to a, \to x \times (\to y \times \to z) = \to b, (\to x \times \to y) \times \to z = \to c

\to a \cdot \to x = \frac{3}{2}, \to a \cdot \to y = \frac{7}{4}

and

| \to a | = 2

.

Then which of the following option(s) is/are CORRECT ?

Q. Solve the following equations:

2 x - 4 y + 9 z = 28

7 x + 3 y - 5 z = 3

9 x + 10 y - 11 z = 4

(i) If a(y+z) = b(z+x) = c(x+y) and out of a, b, c no two of them are equal then show that,

\frac{y - z}{a (b - c)} = \frac{z - x}{b (c - a)} = \frac{x - y}{c (a - b)}

(ii) If

\frac{x}{3 x - y - z} = \frac{y}{3 y - z - x} = \frac{z}{3 z - x - y}

and

x + y + z \neq 0

then show that the value of each ratio is equal to 1.

(iii) If

\frac{a x + b y}{x + y} = \frac{b x + a z}{x + z} = \frac{a y + b z}{y + z}

and

x + y + z \neq 0

then show that

\frac{a + b}{2}

.

(iv) If

\frac{y + z}{a} = \frac{z + x}{b} = \frac{x + y}{c}

then show that

\frac{x}{b + c - a} = \frac{y}{c + a - b} = \frac{z}{a + b - c}

.

(v) If

\frac{3 x - 5 y}{5 z + 3 y} = \frac{x + 5 z}{y - 5 x} = \frac{y - z}{x - z}

then show that every ratio =

\frac{x}{y}

Q. If

x, y, z \in R^{+}

and

a, b, c

are non-zero such that

∣ ∣ ∣ ∣ \begin{matrix} a x & c y & b z b y & a z & c x c z & b x & a y \end{matrix} ∣ ∣ ∣ ∣ = ∣ ∣ ∣ ∣ \begin{matrix} a x & b x & c x c y & a y & b y b z & c z & a z \end{matrix} ∣ ∣ ∣ ∣,

then which of the following is/are correct?

Join BYJU'S Learning Program

Let →x,→y and →z be unit vectors such that →x+→y+→z=→a, →x×(→y×→z)=→b, (→x×→y)×→z=→c, →a⋅→x=32, →a⋅→y=74 and |→a|=2. Then which of the following option(s) is/are CORRECT ?