Question

Find the scalar and vector product of two vectors a =3i-4j+5k and b=-2i+j-3k

Answer 1

We have been given two vectors:

$a=3\hat i-4\hat j+5\hat k$

$b=-2\hat i+\hat j-3\hat k$

To find the scalar product of two vectors we will use:

$a.b=a_xb_x+a_yb_y+a_zc_z$

Where, i, j, and k are the components of unit vectors along 'x', 'y', and 'z'.

$a_x=3,b_x=-2, a_y=-4,b_y=1, a_z=5,b_z=-3$

so the scalar product is given by:

$3 \times(-2) + (-4)\times1 + 5\times(-3) = -6 - 4 - 15 = -25$

So the scalar product of vectors a and b is $-25$.

Vector product of the vectors 'a' and 'b' is given by:

$(a_x, b_x, a_z)\times(a_y,b_y,b_z)=(a_y.b_z-a_z.b_y)\hat i+(a_xb_z-b_xa_z)\hat j+(a_xb_y-a_yb_x) \hat k$

Substituting the values of each component we get:

$\begin{pmatrix}3&-4&5\end{pmatrix}\times \begin{pmatrix}-2&1&-3\end{pmatrix}=(7 -1 -5)$

The vector product of 'a' and 'b' is represented by :

$7\hat i-\hat j-5\hat k$

Answer 2

Answer:

Step-by-step explanation:

We have been given two vectors:

a⋅b=(3  

i

^

−4  

j

^

​

+5  

k

^

)⋅(−2  

i

^

+  

j

^

​

−3  

k

^

)

=−6−4−15

=−25

a×b=  

∣

∣

∣

∣

∣

∣

∣

∣

​

 

i

^

 

3

−2

​

 

j

^

​

 

−4

1

​

 

k

^

 

5

−3

​

 

∣

∣

∣

∣

∣

∣

∣

∣

​

 =7  

i

^

−  

j

^

​

−5  

k

^

 

Note b×a=−7  

i

^

+  

j

^

​

+5  

k

^

To find the scalar product of two vectors we will use:

Where, i, j, and k are the components of unit vectors along 'x', 'y', and 'z'.

so the scalar product is given by:

So the scalar product of vectors a and b is .

Vector product of the vectors 'a' and 'b' is given by:

Substituting the values of each component we get:

The vector product of 'a' and 'b' is represented by :