Each of two vectors, $\overrightarrow { \mathrm { D } } _ { 1 }$ and $\overrightarrow { \mathrm { D } } _ { 2 }$ ,lies along a coordinate axis in the xy plane.Each vector has its tail at the origin,and the dot product of the two vectors is $\overrightarrow { \mathrm { D } } _ { 1 } \cdot \overrightarrow { \mathrm { D } } _ { 2 } = - \left| \overrightarrow { \mathrm { D } } _ { 1 } \right| \left| \overrightarrow { \mathrm { D } } _ { 2 } \right|$ .Which possibility is correct?
A) $$\overrightarrow { \mathrm { D } } _ { 1 }$$ and $$\overrightarrow { \mathrm { D } } _ { 2 }$$ both lie along the positive x axis.
B) $\overrightarrow { \mathrm { D } } _ { 1 }$ lies along the positive x axis. $\overrightarrow { \mathrm { D } } _ { 2 }$ lies along the negative x axis.
C) $\overrightarrow { \mathrm { D } } _ { 1 }$ and $\overrightarrow { \mathrm { D } } _ { 2 }$ both lie along the positive y axis.
D) $\overrightarrow { \mathrm { D } } _ { 1 }$ lies along the negative x axis. $\overrightarrow { \mathrm { D } } _ { 2 }$ lies along the negative y axis.
E) $\overrightarrow { \mathrm { D } } _ { 1 }$ lies along the positive y axis. $\overrightarrow { \mathrm { D } } _ { 2 }$ lies along the negative x axis.

Question

Quizplus · Accepted Answer

Since both vectors lie on the same axis, their dot product is equal to the product of their magnitudes.
Let the two vectors be a and b, where a lies along the x-axis and b lies along either the positive or negative x-axis.
So, |a|*|b| = a.b = 11ea99f6_d745_77c7_8bb6_a7a196f9c600_TB5764_11_TB5764_11_TB5764_11_TB5764_11_TB5764_11_TB5764_11_TB5764_11_TB5764_11.
But, |a| and |b| are non-negative scalars. So, their product must also be non-negative. But, the given dot product is negative, which is not possible.
Hence, the only possibility is that b lies along the negative x-axis and hence, a lies along the positive x-axis. So, option B is correct.

Each of Two Vectors D→1\overrightarrow { \mathrm { D } } _ { 1 }D1​

Each of Two Vectors $\overrightarrow { \mathrm { D } } _ { 1 }$