If a - i - a - i - j - a - i - j - k -1- then a -a 0 -b i -c j -d i - j - k -

(b) i^ Let #160;a #8594;=a1i^+a2j^+a3k^a #8594;. #160;i^=a1and #160;a #8594;. #160;i^+j^=a1+a2and #160;a #8594;. #160;i^+j^+k^=a1+a2+a3Giv ...

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Byju's Answer

Standard XII

Mathematics

Definite Integral as Limit of Sum

If a →· i ∧= ...

Question

If $\vec{a} \cdot \hat{i} = \vec{a} \cdot (\hat{i} + \hat{j}) = \vec{a} \cdot (\hat{i} + \hat{j} + \hat{k}) = 1,$ then $\vec{a} =$
(a) $\vec{0}$

(b) $\hat{i}$

(c) $\hat{j}$

(d) $\hat{i} + \hat{j} + \hat{k}$

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Solution

(b) $\hat{i}$

$Let \vec{a} = a_{1} \hat{i} + a_{2} \hat{j} + a_{3} \hat{k} \vec{a} . \hat{i} = a_{1} and \vec{a} . (\hat{i} + \hat{j}) = a_{1} + a_{2} and \vec{a} . (\hat{i} + \hat{j} + \hat{k}) = a_{1} + a_{2} + a_{3} Given, \vec{a} . \hat{i} = \vec{a} . (\hat{i} + \hat{j}) = \vec{a} . (\hat{i} + \hat{j} + \hat{k}) = 1 \Rightarrow a_{1} = a_{1} + a_{2} = a_{1} + a_{2} + a_{3} = 1 \Rightarrow a_{1} = 1, a_{2} = 0, a_{3} = 0 So, \vec{a} = a_{1} \hat{i} + a_{2} \hat{j} + a_{3} \hat{k} = 1 \hat{i} + 0 \hat{j} + 0 \hat{k} = \hat{i}$

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Similar questions

Let $\to a = i + 2 j + k, \to b = i - j + k a n d \to c = i + j - k,$ a vector in the plane $\to a a n d \to b$ whose projection on $\to c i s \frac{1}{\sqrt{3}} i s . . . . . . . .$

Q. A unit vector perpendicular to both

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

is

(a)

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

(b)

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

(c)

\frac{1}{\sqrt{3}} (\hat{i} + \hat{j} + \hat{k})

(d)

\frac{1}{\sqrt{3}} (\hat{i} - \hat{j} + \hat{k})

Q. Find

(\to a \times \to b) . (\to c \times \to d)

\to a = \to i + \to j + \to k, \to b = 2 \to i + \to k, \to c = 2 \to i + \to j + \to k

and

\to d = \to i + \to j + 2 \to k

Q. Assertion :If

i \times [(a - j) \times i] + j \times [(a - k) \times j] + k [(a - i) \times k] = 0

, then

a = \frac{1}{2} (i + j + k)

. Reason:

(a . i) i + (a . j) j + (a . k) k = a

for any vector

a

Q. Consider the following C-function in which a [n] and b [m] are two sorted integer arrays and c [n+m] be another integer array.

void xyz (int a[ ], int b[ ], int c[ ])
{
int i, j, k;
i = j = k = 0;
while ((i < n) && (j < m))
if (a[i] < b[j])
c[k++] = a[i ++];
else
c[k++] = b[j ++];
}

Which of the following condition(s) hold after the termination of the while loop?

(i) j < m, k = n + j - 1 & a[n- 1] < b[j] if i = n
(ii) i < n, k = m + i - 1 & b[m- 1]

\leq

a[i] if j = m

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If a →·i^=a →·i^+j^=a →·i^+j^+k^=1, then a →= (a) 0 → (b) i^ (c) j^ (d) i^+j^+k^

(b) i^ Let a→=a1i^+a2j^+a3k^a→. i^=a1and a→. i^+j^=a1+a2and a→. i^+j^+k^=a1+a2+a3Given,a→. i^=a→. i^+j^=a→. i^+j^+k^=1⇒a1=a1+a2=a1+a2+a3=1⇒a1=1, a2=0, a3=0So, a→=a1i^+a2j^+a3k^=1 i^+0j^+0k^=i^

If $\vec{a} \cdot \hat{i} = \vec{a} \cdot (\hat{i} + \hat{j}) = \vec{a} \cdot (\hat{i} + \hat{j} + \hat{k}) = 1,$ then $\vec{a} =$
(a) $\vec{0}$

(b) $\hat{i}$

(c) $\hat{j}$

(d) $\hat{i} + \hat{j} + \hat{k}$